Trees

Steven J. Zeil

Last modified: Jul 16, 2014

Contents:

1. Tree Terminology

2. Tree Traversal

2.1 Recursive Traversals

3. Example: Processing Expressions

4. Example: Processing XML

5. Using Trees for Searching

5.1 How Fast Are Binary Search Trees?

Most of the data structures we have looked at so far have been devoted to keeping a collection of elements in some linear order.

Trees

Trees are the most common non-linear data structure in computer science.

Useful in representing things that naturally occur in hierarchies
- (e.g., many company organization charts are trees)

and for things that are related in a “is-composed-of” or “contains” manner
- (e.g., this country is composed of states, each state is composed of counties, each county contains cities, each city contains streets, etc.)

Trees also turn out to be exceedingly useful in implementing fast searching and insertion.

Properly implemented, a tree can be both searched and inserted into in O(log N) time.

1. Tree Terminology

Definition

A tree is a collection of nodes.

If nonempty, the collection includes a designated node r, the root, and zero or more (sub)trees T1, T2, … , Tk, each of whose roots are connected by an edge to r.

A Tree

We can designate A as the root, and we note that
the collections of nodes {B}, {C, F}, {D}, and {E,G,H,I}, together with their edges, are all trees whose own roots are connected to A.

It’s a subtle, but important point to note that in discussing trees, we sometimes focus on the things connected to the root as individual nodes, and other times as entire trees.

Botanical Families

Each node except the root has a parent.
Parent nodes have children. Nodes without children are leaves.
Nodes with the same parent are siblings.

Binary and General Trees

A tree in which every parent has at most 2 children is a binary tree.
Trees in which parents may have more than 2 children are general trees.

2. Tree Traversal

Tree Traversal

Many algorithms for manipulating trees need to “traverse” the tree, to visit each node in the tree and process the data in that node. In this section, we’ll look at some prototype algorithms for traversing trees.

All of these use recursion, because trees are easiest to handle that way.

Kinds of Traversals

A pre-order traversal is one in which the data of each node is processed before visiting any of its children.
A post-order traversal is one in which the data of each node is processed after visiting all of its children.
An in-order traversal is one in which the data of each node is processed after visiting its left child but before visiting its right child.
- This traversal is specific to binary trees.

Example: Expression Trees

Constants and variables go in the leaves,
Each non-leaf node represents the application of some operator to the subtrees representing its operands.

The tree here, for example, shows the product of a sum and of a subtraction.

Traversing an Expression Tree

Pre-order:

* + 13 a - x 1

In-order:

13 + a * x - 1

Compare to ((13+a)*(x–1)), the “natural” way to write this expression.

Post-order:

13 a + x 1 - *

Post-order traversal yields post-fix notation, which in turn is related to algorithms for expression evaluation.

2.1 Recursive Traversals

Recursive Traversals

Let’s suppose that we have a binary tree whose nodes are declared as shown here.

struct BinaryTreeNode 
{
   T data;
   BinaryTreeNode* left;
   BinaryTreeNode* right;
};

It’s fairly easy to write pre-, in-, and post-order traversal algorithms using recursion.

A Basic Traversal

void basicTraverse (BinaryTreeNode* t)
{
  if (t != 0)
    {
      basicTraverse(t->left);
      basicTraverse(t->right);
    }
}

This is the basic structure for a recursive traversal. If this function is called with a null pointer, we do nothing. But if we have a real pointer to some node, we invoke the function recursively on the left and right subtrees. In this manner, we will eventually visit every node in the tree. The problem is, this basic form doesn’t do anything.

Pre-Order Traversals

But we can convert it into a pre-order traversal by applying the rule:

process the node before visiting its children


void preOrder (BinaryTreeNode* t)
{
  if (t != 0)
    {
      foo (t->data);
      preOrder(t->left);
      preOrder(t->right);
    }
}

Post-Order Traversals


void postOrder (BinaryTreeNode* t)
{
  if (t != 0)
    {
      postOrder(t->left);
      postOrder(t->right);
      foo (t->data);
    }
}

We get a post-order traversal by applying the rule:

process the node after visiting its children

In-Order Traversals

And we get an in-order traversal by applying the rule:

process the node after visiting its left descendents and before visiting its right descendents.


void inOrder (BinaryTreeNode* t)
{
  if (t != 0)
    {
      inOrder(t->Left);
      foo (t->Element);
      inOrder(t->Right);
    }
}

Demo

Run the different traversals

3. Example: Processing Expressions

Example: Processing Expressions

We’ll develop a program that can simplify certain basic arithmetic expressions, e.g., converting (2 - 2) * x + 1 * (y + z) to y + z.

The Data Structure

class Expression {
private:
  std::string opOrVarName;
  bool thisIsAConstant;
  bool thisIsAVariable;
  Expression* left;
  Expression* right;
  int value;
   ⋮

Note the left and right pointers, which give this it’s essential “tree-ness”.

The Application

simplifier.cpp

#include <iostream>

#include "expression.h"

using namespace std;




int main(int argc, char** argv)
{
  Expression expr;
  cin >> expr;
  cout << "Input: " << expr << endl;
  expr.normalize();
  cout << "Normalized: " << expr << endl;

  return 0;
}

expression.h

#ifndef EXPRESSION_H
#define EXPRESSION_H

#include <iostream>
#include <string>


class Expression {
private:
  std::string opOrVarName;
  bool thisIsAConstant;
  bool thisIsAVariable;
  Expression* left;
  Expression* right;
  int value;

public:
  // Create a reference to a variable
  Expression (std::string variableName);

  // Create a (binary) operator with two operands
  Expression (std::string op,
          const Expression& left,
          const Expression& right);

  // Create a constant expression
  Expression (int value = 0);

  // Big 3
  Expression (const Expression& expr);
  Expression& operator= (const Expression& expr);

  ~Expression();


  // Attributes

  bool isAConstant() const {return thisIsAConstant;}
  bool isAVariable() const {return thisIsAVariable;}

  int getValue() const  {return value;}
  //pre: isAConstant()

  std::string getVariableName() const {return opOrVarName;}
  //pre: isAVariable()


  std::string getOperator() const  {return opOrVarName;}
  //pre: !isAVariable() && !isAConstant()


  // Sub-expressions

  Expression* getLeft () const {return left;}
  //pre: !isAVariable() && !isAConstant()

  Expression* getRight () const {return right;}
  //pre: !isAVariable() && !isAConstant()


  // Manipulating

  // Apply the distributive laws a*(b+c) => a*b + a*c
  //   and a*(b-c) => a*b - a*c whenever possible
  void distribute();

  // Perform trivial simplifications: 0*a => 0, 1*a => a, a/1 => a,
  //    a+0 => a, a-0 => a  c1 op c2 => c3  where a is any expression,
  //    the ci are constants, and op is any of * / + -
  void simplify();

  // Reduce an expression to normalized form
  void normalize()
  {
    distribute();
    simplify();
  }

private:
  bool expression (std::istream& input);
  bool product (std::istream& input);
  bool term (std::istream& input);

  friend std::istream& operator>> (std::istream& in, Expression& exp);
  friend std::ostream& operator<< (std::ostream& out, const Expression& exp);

};


std::ostream& operator<< (std::ostream& out, const Expression& exp);
std::istream& operator>> (std::istream& in, Expression& exp);




#endif

expression.cpp

#include "expression.h"
#include <cstdlib>

using namespace std;


// Create a reference to a variable
Expression::Expression (std::string variableName)
  : thisIsAConstant(false), thisIsAVariable(true),
    left(NULL), right(NULL), opOrVarName(variableName),
    value(-12345)
{
}

// Create a (binary) operator with two operands
Expression::Expression (std::string op,
            const Expression& leftOpnd,
            const Expression& rightOpnd)
  : thisIsAConstant(false), thisIsAVariable(false),
    opOrVarName(op),
    value(-12345)
{
  left = new Expression(leftOpnd);
  right = new Expression(rightOpnd);
}

// Create a constant expression
Expression::Expression (int val)
  : thisIsAConstant(true), thisIsAVariable(false),
    left(NULL), right(NULL), opOrVarName(""),
    value(val)
{
}





// Big 3
Expression::Expression (const Expression& expr)
  : thisIsAConstant(expr.thisIsAConstant),
    thisIsAVariable(expr.thisIsAVariable),
    opOrVarName(expr.opOrVarName),
    value(expr.value)
{
    if (expr.left != NULL)
       left = new Expression(*(expr.left));
    else
       left = NULL;
    if (expr.right != NULL)
       right = new Expression(*(expr.right));
    else
       right = NULL;
}



Expression& Expression::operator= (const Expression& expr)
{
  if (this != &expr) {
    Expression* oldLeft = left;
    Expression* oldRight = right;

    thisIsAConstant = expr.thisIsAConstant;
    thisIsAVariable = expr.thisIsAVariable;
    opOrVarName= expr.opOrVarName;
    value = expr.value;
    if (expr.left != NULL)
       left = new Expression(*(expr.left));
    else
       left = NULL;
    if (expr.right != NULL)
       right = new Expression(*(expr.right));
    else
       right = NULL;

    delete oldLeft;
    delete oldRight;
  }
  return *this;
}


Expression::~Expression()
{
  delete left;
  delete right;
}




// Manipulating

// Apply the distributive laws a*(b+c) => a*b + a*c
//   and a*(b-c) => a*b - a*c whenever possible
void Expression::distribute()
{
  if ((!thisIsAVariable) && (!thisIsAConstant))
    {
      left->distribute();
      right->distribute();
      if (opOrVarName == "*")
    {
      if (left->opOrVarName == "+" || left->opOrVarName == "-")
        {
          // (a + b) * c
          Expression* a = left->left;
          Expression* b = left->right;
          Expression* c = right;
          *this = Expression(left->opOrVarName,
                 Expression ("*", *a, *c),
                 Expression ("*", *b, *c));
          distribute();
        }
      else if (right->opOrVarName == "+" || right->opOrVarName == "-")
        {
          // a * (b + c)
          Expression* a = left;
          Expression* b = right->left;
          Expression* c = right->right;
          *this = Expression(right->opOrVarName,
                 Expression ("*", *a, *b),
                 Expression ("*", *a, *c));
          distribute();
        }
    }
    }
}

// Perform trivial simplifications: 0*a => 0, 1*a => a, a/1 => a,
//    a+0 => a, a-0 => a  c1 op c2 => c3  where a is any expression,
//    the ci are constants, and op is any of * / + -
void Expression::simplify()
{
  if ((!thisIsAVariable) && (!thisIsAConstant))
    {
      left->simplify();
      right->simplify();

      if (left->isAConstant() && right->isAConstant())
    {
      // Both operands are constants, evaluate the operation
      // and change this expression to a constant for the result.
      if (opOrVarName == "+")
        value = left->value + right->value;
      else if (opOrVarName == "-")
        value = left->value - right->value;
      else if (opOrVarName == "*")
        value = left->value * right->value;
      else if (opOrVarName == "/")
        value = left->value / right->value;
      thisIsAConstant = true;
      opOrVarName = "";
      delete left;
      delete right;
      left = right = NULL;
    }
      else if (left->isAConstant())
    {
      if (opOrVarName == "+" && left->value == 0)
        {
          *this = *right;
        }
      else if (opOrVarName == "*" && left->value == 0)
        {
          *this = Expression(0);
        }
      else if (opOrVarName == "*" && left->value == 1)
        {
          *this = *right;
        }
    }
      else if (right->isAConstant())
    {
      if (opOrVarName == "+" && right->value == 0)
        {
          *this = *left;
        }
      else if (opOrVarName == "-" && right->value == 0)
        {
          *this = *left;
        }
      else if (opOrVarName == "*" && right->value == 1)
        {
          *this = *left;
        }
      else if (opOrVarName == "*" && right->value == 0)
        {
          *this = Expression(0);
        }
      else if (opOrVarName == "/" && right->value == 0)
        {
          *this = Expression(0);
        }
    }
    }
}



std::ostream& operator<< (std::ostream& out, const Expression& exp)
{
  if (exp.isAConstant())
    out << exp.getValue();
  else if (exp.isAVariable())
    out << exp.getVariableName();
  else
    {
      if (!exp.left->isAVariable() &&!exp.left->isAConstant())
    out << "(" << *(exp.left) << ")";
      else
    out << *(exp.left);
      out << exp.getOperator();
      if (!exp.right->isAVariable() &&!exp.right->isAConstant())
    out << "(" << *(exp.right) << ")";
      else
    out << *(exp.right);
    }
  return out;
}


std::istream& operator>> (std::istream& in, Expression& exp)
{
  bool ok = exp.expression (in);
  if (!ok)
    in.setstate(ios::failbit);
  return in;
}


bool Expression::expression (istream& input)
{
  bool OK = product (input);

  char c = ' ';
  while (OK && input >> c)
    {
      if (c == '+' || c == '-')
    {
      string op (1, c);
      Expression right;
      OK = right.product(input);
      if (OK)
        *this = Expression(op, *this, right);
    }
      else
    {
      input.putback(c);
      return OK;
    }
    }
}


bool Expression::product (istream& input)
{
  bool OK = term(input);

  char c = ' ';
  while (OK && input >> c)
    {
      if (c == '*' || c == '/')
    {
      string op (1, c);
      Expression right;
      OK = right.term(input);
      if (OK)
        *this = Expression(op, *this, right);
    }
      else
    {
      input.putback(c);
      return OK;
    }
    }
}


bool Expression::term (istream& input)
{
  char c = ' ';
  input >> c;
  if (c == '(')
    {
      bool OK = expression (input);
      if (OK)
    input >> c;
      return OK && (c == ')');
    }
  else if ((c >= 'A' && c <= 'Z') || (c >= 'a' && c <= 'z'))
    {
      string varName (1, c);
      int cc = input.get();
      while (cc > 0 && (cc >= 'A' && cc <= 'Z') || (cc >= 'a' && cc <= 'z'))
    {
      varName += (char)cc;
      cc = input.get();
    }
      if (cc >= 0)
    input.putback(cc);
      *this = Expression(varName);
      return true;
    }
  else if (c >= '0' && c <= '9')
    {
      input.putback(c);
      int val;
      input >> val;
      *this = Expression(val);
      return true;
    }
  else
    {
      return false;
    }
}

We’ll look at selected parts in more detail.

Printing Expressions

printExpr.cpp

std::ostream& operator<< (std::ostream& out, const Expression& exp)
{
  if (exp.isAConstant())
    out << exp.getValue();
  else if (exp.isAVariable())
    out << exp.getVariableName();
  else
    {
      if (!exp.left->isAVariable() &&!exp.left->isAConstant())
        out << "(" << *(exp.left) << ")";
      else
        out << *(exp.left);
      out << exp.getOperator();
      if (!exp.right->isAVariable() &&!exp.right->isAConstant())
        out << "(" << *(exp.right) << ")";
      else
        out << *(exp.right);
    }
  return out;
}

Distributive Law

distrib.cpp

// Apply the distributive laws a*(b+c) => a*b + a*c
//   and a*(b-c) => a*b - a*c whenever possible
void Expression::distribute()
{
  if ((!thisIsAVariable) && (!thisIsAConstant))
    {
      left->distribute();
      right->distribute();
      if (opOrVarName == "*")
        {
          if (left->opOrVarName == "+" || left->opOrVarName == "-")
            {
              // (a + b) * c
              Expression* a = left->left;
              Expression* b = left->right;
              Expression* c = right;
              *this = Expression(left->opOrVarName,
                                 Expression ("*", *a, *c),
                                 Expression ("*", *b, *c));
              distribute();
            }
          else if (right->opOrVarName == "+" || right->opOrVarName == "-")
            {
              // a * (b + c)
              Expression* a = left;
              Expression* b = right->left;
              Expression* c = right->right;
              *this = Expression(right->opOrVarName,
                                 Expression ("*", *a, *b),
                                 Expression ("*", *a, *c));
              distribute();
            }
        }
    }
}

Simplifying Expressions

simplify.cpp

// Perform trivial simplifications: 0*a => 0, 1*a => a, a/1 => a,
//    a+0 => a, a-0 => a  c1 op c2 => c3  where a is any expression,
//    the ci are constants, and op is any of * / + -
void Expression::simplify()
{
  if ((!thisIsAVariable) && (!thisIsAConstant))
    {
      left->simplify();
      right->simplify();

      if (left->isAConstant() && right->isAConstant())
        {
          // Both operands are constants, evaluate the operation
          // and change this expression to a constant for the result.
          if (opOrVarName == "+")
            value = left->value + right->value;
          else if (opOrVarName == "-")
            value = left->value - right->value;
          else if (opOrVarName == "*")
            value = left->value * right->value;
          else if (opOrVarName == "/")
            value = left->value / right->value;
          thisIsAConstant = true;
          opOrVarName = "";
          delete left;
          delete right;
          left = right = NULL;
        }
      else if (left->isAConstant())
        {
          if (opOrVarName == "+" && left->value == 0)
            {
              *this = *right;
            }
          else if (opOrVarName == "*" && left->value == 0)
            {
              *this = Expression(0);
            }
          else if (opOrVarName == "*" && left->value == 1)
            {
              *this = *right;
            }
        }
      else if (right->isAConstant())
        {
          if (opOrVarName == "+" && right->value == 0)
            {
              *this = *left;
            }
          else if (opOrVarName == "-" && right->value == 0)
            {
              *this = *left;
            }
          else if (opOrVarName == "*" && right->value == 1)
            {
              *this = *left;
            }
          else if (opOrVarName == "*" && right->value == 0)
            {
              *this = Expression(0);
            }
          else if (opOrVarName == "/" && right->value == 0)
            {
              *this = Expression(0);
            }
        }
    }
}

4. Example: Processing XML

XML

XML is a markup language used for exchanging data among a wide variety of programs on the web. It is flexible enough to represent almost any kind of data.

In XML, data is described by structures consisting of nested “elements” and text.

An element is described as an opening “tag”, the contents of the element, and a closing tag.

XML Tags

Tags are written inside < > brackets. Inside the brackets, an opening tag has a tag name followed by zero or more “attributes”. An attribute is a name="value" pair, with the value inside quotes. Closing tags have no attributes, and are indicated by placing the tag name immediately after a ‘/’ character.
Finally, if an element has no text and no internal elements, it can be written as <tag attributes> </tag> or abbreviated as <tag attributes />.

An example of XML:

<Question type="Choice" id="ultimate">
<QCategory>trial</QCategory>
<Body>What is the answer to the ultimate question of life, 
the universe, and everything?</Body>
<Choices>
<Choice>a good nap</Choice>
<Choice value="1">42</Choice>
<Choice>inner peace</Choice>
<Choice>money</Choice>
</Choices>
<AnswerKey>42</AnswerKey>
<Explanation>D Adams,
The Hitchhiker's Guide to the Galaxy</Explanation>
<Revisions>5/15/2001 1:35:29 PM</Revisions>
</Question>

XML and Trees

Although it may not be obvious, XML actually describes a tree structure.

For example, the structure above shows that all the elements are inside a “Question”. One of those elements inside the Question is a “Choices” element, and each individual “Choice” occurs inside there.We can diagram this tree structure as shown here.

XML is closely related to HTML

HTML allows many attributes to be written without placing the value in quotes,
empty elements can be written as <tagname> instead of <tagname/>,
and some non-empty elements can be written without a closing </tagname>

Just a test

Nothing much to see here.

Move along.

For example, the text shown here can be produced by the following XML-legal HTML:

<html>
  <head>
    <title>Just a Test</title>
  </head>

  <body>
    <h1>Just a test</h1>
<p>Nothing much to see <a href="test.html">here</a>.
</p>
<hr/>
<p>
Move <a href="nextpage.html">along</a>.
</p>
  </body>
</html>

The tree structure for this HTML page is:

A program that would read a web page and print a list of all links (<a> elements with href= attributes).

/** Example of tree manipulation using XML documents */

#include <iostream>

using namespace std;

#include <xercesc/parsers/XercesDOMParser.hpp>
#include <xercesc/dom/DOM.hpp>
#include <xercesc/sax/HandlerBase.hpp>
#include <xercesc/util/XMLString.hpp>
#include <xercesc/util/PlatformUtils.hpp>

using namespace XERCES_CPP_NAMESPACE;

DOMDocument* readXML (const char *xmlFile) 
{
  ⋮
}

string getHrefAttribute (DOMNode* linkNode)
{
  DOMElement* linkNodeE = (DOMElement*)linkNode;
  const XMLCh* href = XMLString::transcode("href");
  const XMLCh* attributeValue = linkNodeE->getAttribute(href);
  return string(XMLString::transcode(attributeValue));
}


void processTree (DOMNode *tree)
{
  if (tree != NULL)
    {
      if (tree->getNodeType() == DOMNode::ELEMENT_NODE)
        {
          const XMLCh* elName = tree->getNodeName();
          const XMLCh* aName = XMLString::transcode("a");
          if (XMLString::equals(elName, aName))
            cout << "Link to " << getHrefAttribute(tree) << endl;
        }
      for (DOMNode* child = tree->getFirstChild();
           child != NULL; child = child->getNextSibling())
        processTree(child);
    }
}


int main(int argc, char **argv) 
{
  if (argc != 2)
    {
      cerr << "usage: " << argv[0] << " xmlfile" << endl;
      return 1;
    }

  // Initialize the Xerces XML C++ library
  try {
    XMLPlatformUtils::Initialize();
  }
  catch (const XMLException& toCatch) {
    char* message = XMLString::transcode(toCatch.getMessage());
    cout << "Error during initialization! :\n"
         << message << "\n";
    XMLString::release(&message);
    return 1;
  }

  DOMDocument* doc = readXML(argv[1]);
  if (doc == 0)
    {
      cerr << "Could not read " << argv[1] << endl;
      return 2;
    }
  processTree (doc->getDocumentElement());

  // Cleanup
  doc->release();
  return 0;
}

based upon the Xerces-C++ library.

If this code is compiled and run on the HTML page we have just seen, it would print

Link to test.html
Link to nextpage.html

Processing XML as a Tree

void processTree (DOMNode *tree)
{
  if (tree != NULL)
    {
      if (tree->getNodeType() == DOMNode::ELEMENT_NODE)
        {
          const XMLCh* elName = tree->getNodeName();
          const XMLCh* aName = XMLString::transcode("a");
          if (XMLString::equals(elName, aName))
            cout << "Link to " << getHrefAttribute(tree) << endl;
        }
      for (DOMNode* child = tree->getFirstChild();
           child != NULL; child = child->getNextSibling())
        processTree(child);
    }
}

Now, this code features an interface that you have never seen before, and a lot of the details are bound to look mysterious.

Nonetheless, if you look at where the processTree function calls itself, you can readily tell that this function works by pre-order traversal.

5. Using Trees for Searching

Search Trees

For large collections of data, our current data structure allow us to do fast searches or fast insertions, but not both.

Arrays and vectors allow fast searching (binary search)
- But inserting an element requires, on average, moving half the elements.
Linked lists allow rapid insertions
- But searches must be sequential
- Looks at half the elements, on average

Trees offer us a way out of this conflict.

Definition: Binary Search Trees

A tree in which every parent has at most 2 children is a binary tree.

binary search tree\(T_{L}\)\(T_{R}\)

The value in n is greater than the values in every node in \(T_{L}\).
The value in n is less than the values in every node in \(T_{R}\).
Both \(T_{L}\) and \(T_{R}\) are binary search trees.

The Binary Search Tree ADT

Let’s look at the basic interface for a binary search tree.

template <class T>
class node 
{
public:
  node (const T & v, 
        node<T> * left, 
        node<T> * right)
    : value(v), 
      leftChild(left), 
      rightChild(right) { }
  
  T value;
  node<T> * leftChild;
  node<T> * rightChild;
};

The constructor initializes the data (value) field and the two child pointers.

Searching a Binary Tree

template <class T> 
node<T>* find (const T& element, 
               const node<T>* t)
{ 
 if (t == NULL)
   return NULL;
 if (element < t->value)
   return find(element, t->left);
 else if (t->value < element)
   return find(element, t->right);
 else  // t->value == element
   return t;
}

We search a tree by comparing the value we’re searching for to the “current” node.

If the value we want is smaller, we look in the left subtree.
If the value we want is larger, we look in the right subtree.

Demo

Run this algorithm.

Inserting into Binary Search Trees

template <class T> 
void insert (const T& element, node<T>*& t)
{ 
 if (t == NULL)
   t = new node<T>(element, NULL, NULL);
 if (element < t->value)
   insert (element, t->left);
 else if (t->value < element)
   insert (element, t->right);
 else  // t->value == element
   return; // If we want no duplicates
   // insert (element, t->right); // If we permit duplicates
}

Similar to the search function, we move left and right by comparing the data to the current node t
Only difference is what happens when we reach a pointer that is null

Demo

Run this algorithm.

5.1 How Fast Are Binary Search Trees?

Trees Can Be Fast

Each step in the BST insert and find algorithms move one level deeper in the tree.

The number of recursive calls/loop iterations in all these algorithms is therefore no greater than the height of the tree.
But how high can a BST be?

That depends on how well the tree is “balanced”.

Shapes of Trees

What Determines a Tree’s Shape?

The shape of the tree depends upon the order of insertions.

The worst case is when the data being inserted is already in order (or in reverse order).
- In that case, the tree degenerates into a sorted linked list, as shown earlier.

The best case is when the tree is balanced, meaning that, for each node, the heights of the node’s children are nearly the same.
- Happens (approximately) when data is randomly ordered before being inserted