Rooted Trees

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Rooted Trees - PowerPoint PPT Presentation

Rooted Trees. More definitions. root. internal vertex. descendants of g. ancestor of d. leaf. parent of d. child of c. subtree. sibling of d.

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More definitions

root

internal vertex

descendants of g

ancestor of d

leaf

parent of d

child of c

subtree

sibling of d

Definition 2. A rooted tree is called an m-ary tree if every internal vertex has no more than m children. The tree is called a full m-ary tree if every internal vertex has exactly m children. An m-ary tree with m = 2 is called a binary tree.

Are these full m-ary trees?

Properties of trees

Theorem 2. A tree with n vertices has n - 1 edges

• Choose root, r.
• Set up one-to-one correspondence between edges and vertices other than r.
• There are n – 1 vertices so there are n – 1 edges.

Theorem 3. A full m-ary tree with i internal vertices contains n = mi + 1 vertices

• Every vertex (except root) is the child of an internal vertex.
• Each of the i internal vertices has m children.
• There are mi vertices (other than the root).
• Therefore n = mi + 1.

i = 4 internal vertices

m = 3

n = 3 × 4 + 1 = 13

Theorem 4.A full m-ary tree with

• n vertices has i = (n – 1)/m internal vertices and l = [(m – 1)n + 1]/m leaves
• i internal vertices has n = mi + 1 vertices and l = (m – 1)i + 1 leaves
• l leaves has n = (ml – 1)/(m – 1) vertices and i = (l – 1)/(m – 1) internal vertices

Corollary 1. If an m-ary tree of height h has l leaves, then h logml. If the m-ary tree is full and balanced, then h = logml.