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1 Chapter 10 Trees

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Tree Definition 1. A tree is a connected undirected graph with no simple circuits. Theorem 1. An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices. 2

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Which graphs are trees? 3 a) b) c)

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Specify a vertex as root 4 c) Then, direct each edge away from the root. ROOT

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Specify a root. 5 a) ROOT Then, direct each edge away from the root.

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Specify a root. 6 a) ROOT Then, direct each edge away from the root.

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Specify a root. 7 a) ROOT Then, direct each edge away from the root. A directed graph called a rooted tree results.

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What if a different root is chosen? 8 a) ROOT Then, direct each edge away from the root.

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What if a different root is chosen? 9 a) ROOT Then, direct each edge away from the root.

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What if a different root is chosen? 10 a) ROOT Then, direct each edge away from the root.

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What if a different root is chosen? 11 a) ROOT Then, direct each edge away from the root. A different rooted tree results.

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12 Jake’s Pizza Shop Tree Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len

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13 Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len A Tree Has a Root TREE ROOT

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14 Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len Leaf nodes have no children LEAF NODES

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15 Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len A Tree Has Levels LEVEL 0

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16 Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len Level One LEVEL 1

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17 Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len Level Two LEVEL 2

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18 SIBLINGS Sibling nodes have same parent Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len

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19 SIBLINGS Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len Sibling nodes have same parent

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20 Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len A Subtree LEFT SUBTREE OF ROOT ROOT

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21 Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len Another Subtree RIGHT SUBTREE OF ROOT ROOT

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Internal Vertex A vertex that has children is called an internal vertex. The subtree at vertex v is the subgraph of the tree consisting of vertex v and its descendants and all edges incident to those descendants. 22

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23 Owner Jake Manager Brad Chef Carol WaitressWaiter Cook Helper Joyce Chris Max Len How many internal vertices?

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Binary Tree Definition 2 ’. A rooted tree is called a binary tree if every internal vertex has no more than 2 children. The tree is called a full binary tree if every internal vertex has exactly 2 children. 24

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Ordered Binary Tree Definition 2 ’’. An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered. In an ordered binary tree, the two possible children of a vertex are called the left child and the right child, if they exist. 25

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Tree Properties Theorem 2. A tree with N vertices has N-1 edges. Theorem 5. There are at most 2 H leaves in a binary tree of height H. Corallary. If a binary tree with L leaves is full and balanced, then its height is H = log 2 L . 26

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An Ordered Binary Tree 27 Hal Lou Ken Joe Ted Sue Ed Max

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Parent The parent of a non-root vertex is the unique vertex u with a directed edge from u to v. 28

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What is the parent of Ed? 29 Hal Lou Ken Joe Ted Sue Ed Max

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Leaf A vertex is called a leaf if it has no children. 30

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How many leaves? 31 Hal Lou Ken Joe Ted Sue Ed Max

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Ancestors The ancestors of a non-root vertex are all the vertices in the path from root to this vertex. 32

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How many ancestors of Ken? 33 Hal Lou Ken Joe Ted Sue Ed Max

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Descendants The descendants of vertex v are all the vertices that have v as an ancestor. 34

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How many descendants of Hal? 35 Hal Lou Ken Joe Ted Sue Ed Max

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Level The level of vertex v in a rooted tree is the length of the unique path from the root to v. 36

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What is the level of Ted? 37 Hal Lou Ken Joe Ted Sue Ed Max

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Height The height of a rooted tree is the maximum of the levels of its vertices. 38

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What is the height? 39 Hal Lou Ken Joe Ted Sue Ed Max

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Balanced A rooted binary tree of height H is called balanced if all its leaves are at levels H or H- 1. 40

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Is this binary tree balanced? 41 Hal Lou Ken Joe Ted Sue Ed Max

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