Announcements HW2 available, due 10/04, 11:59p. MT1 10/10, 7-9p. - - PowerPoint PPT Presentation

Announcements –

HW2 available, due 10/04, 11:59p. MT1 10/10, 7-9p.

Trees:

“… most important nonlinear structure[s] in computer science.”

• - Donald Knuth, Art of Computer Programming Vol 1

A tree: _____________________________________ We’ll study more specific trees:

Tree terminology:

• One of the vertices is called the root of the tree. Guess which one it is.
• Make an English word containing the names of the vertices that have a parent but no

sibling.

• How many parents does each vertex have?
• Which vertex has the fewest children?
• Which vertex has the most ancestors? descendants?
• What is d’s depth? What is d’s height?
• List all the vertices is b’s left subtree.
• List all the leaves in the tree.

Tree terminology: (for your reference)

• root: the single node with no parent
• leaf: a node with no children
• child: a node pointed to by me
• parent: the node that points to me
• sibling: another child of my parent
• ancestor: my parent or my parent’s ancestor
• descendent: my child or my child’s descendent
• subtree: a node and its descendents
• depth of node x: number of edges on path from root to x.
• height of node x: number of edges on longest path from x to a leaf.

A rooted tree:

Branching: d-ary trees (binary if d = 2)

A d-ary tree T is either

• OR
• Full d-ary tree:

Perfect d-ary tree: Complete d-ary tree:

Binary Tree Height height(r) -- length of longest path from root r to a leaf

Given a binary tree T, write a recursive defn of the height of T, height(T): Number of nodes in a perfect tree of height h, N(h):

Rooted, directed, ordered, binary trees

Tree ADT: insert remove traverse

1 2 3 4 5 6 7 8 9 10 11 12 13 template<class T> class tree { public: … private: struct Node { T data; Node * left; Node * left; }; Node * root; … };

Theorem: Our implementation of an n item binary tree has ______ null pointers.