Definition ! A (rooted) tree is a structure defined on a finite set - - PDF document

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Trees & Binary Trees

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Definition

! A (rooted) tree is a structure defined

• n a finite set of nodes that either:

– Contains no nodes, or

– Contains a specially designated node called the “root” and k>=0 disjoint sets of nodes T1,…,Tk, where each Ti is a tree. T1,…,Tk are called the subtrees of the root.

! Ordered tree: order of subtrees

matters.

A B E F M L K D G J N C H I

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Treeminology

(Tree Terminology)

! Internal nodes and leaf nodes ! Parent/child relationship ! Siblings: children of the same parent ! Ancestor/descendant relationship ! Degree of a node: number of children

A B E F M L K D G J N C H I

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Treeminology

(Tree Terminology)

! Path: a sequence n1,…,nk of nodes, s.t.

ni is the parent of ni+1.

! Tree height: length (number of edges)

• f longest path from the root to a leaf

! Node depth: length of the path from

the root to the node

A B E F M L K D G J N C H I

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Binary Trees

! A binary tree T is a structure defined

• n a finite set of nodes that either

– Contains no nodes, or – Is comprised of a root node, and two subtrees (left and right) each of which is also a binary tree. A B C F E D G A B C E D G F

≠

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More Terminology

! A full binary tree: each node has either

degree 0 (a leaf) or 2 (has exactly two non-empty children).

! A complete binary tree: a full binary

tree in which all leaves have the same depth.

A B C E D G F A B C E D G F

full: complete:

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Treevia (Tree Facts)

! How many leaf nodes are there in a

complete binary tree of height h?

! What is the number of internal nodes

in such a tree?

! What is the total number of nodes? ! How many leaf nodes are there in a

full binary tree of height h?

! How tall can a full binary tree with n

leaf nodes be?

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Array Implementation

A B C E D G F A B D C E F G

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Array Implementation

class BinaryTreeA { private int size; private Item[] items; // 0..size-1 int leftChild(int i) { return 2*i + 1; } int rightChild(int i) { return 2*i + 2; } int parent(int i) { return floor((i - 1) / 2); } }

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Pointer-Based Implementation

class BinaryTree { private Item item; private BinaryTree left, right; BinaryTree leftChild() { return left; } BinaryTree rightChild() { return right; } boolean isLeaf() { return (left == null && right == nul); } }

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Binary Tree Traversal

! Inorder: visit left subtree, then

the root, then the right subtree.

! Preorder: visit root, left, right. ! Postorder: visit left, right, root.

void inOrder(BinaryTree t) { if (t.leftChild() != null) inOrder(t.leftChild()); doStuff(t); if (t.rightChild() != null) inOrder(r.rightChild()); }