Chapter 10

1. Chapter 10 Trees

2. contents • Introduction • definitions • properties of trees • Applications • binary search trees • Decision trees • Prefix codes • Tree traversal • universal address systems • inorder, preorder and postorder traversal • infix, prefix and postfix notation – linearize(serialize) trees. • trees and sorting -- bubble sort, merge-sort • spanning trees • minimum spanning trees -- Prim's alg; Kruskal's alg.

3. x y ==> X is the parent of Y 10.1 Introduction • A connected graph w/o simple circuits • useful in CS and many fields Def1: A tree is a connected undirected graph w/o simple circuits (cf: Forest) Ex1: Family Tree: Ex2: tree tree has cycle not connected

4. Theorem 1: Theorm1:An undirected graph is a tree iff there is a unique simple path b/t any two of its vertices. Pf:”=>”: T: any tree; x,y: two distinct vertices. Since T is connected, by def., there is a path and hence a simple path a from x to y. Assume there is another simple path b from x to y. Then by a theorem of ch 8 about simple circuit, there is a simple circuit in T.=> T is not a tree, a contradiction! “<=“: There is a path b/t any two vertices ==> T is connected. Assume T contains a simple cycle containing x and y. (i.e., x --a-- y --b-- x ==> T contains two simple paths from x to y, a contradiction.

5. Rooted Tree • A rooted tree is a tree T=(V,E) associated with a distinguished vertex r ∈ V called the root of T. • Given a rooted tree with root r, we can form a directed graph with the property that (u,v) ∈ V2 is an edge of the new directed graph iff u appears immediately before v in the unique simple path from the root r to v.

6. f f g g f g d d d b b b e e e a a a c c c T With root a With root c A tree and Rooted Tree formed by designating two roots

7. Some terminology of rooted tree • T=<V,E,r>: a rooted tree. • The associated digraph of T is a digraph GT = (V,E’) where E’ = {(u,v) ∈ V2|{u,v}∈E and ∃a path r---….---u---v in T} • if (u,v) ∈ E’ => u is called the parent of v (and v a child of u) • Vertices with the same parent are called siblings. • All vertices(except v) in the unique path from the root r to a vertex v are ancestors of v. • v is a descendant of u iff u is an ancestor of v. • Vertices without children are called leaves • Vertices having children are called internal nodes.

8. f g d b e a c subtrees • T=(V,E,r): a rooted tree; a: a vertex. • The subtree of T with a as root is the tree T’=(V’,E’,a) where • V’ = {{a} U {v | v is a descendant of a} • E’ = E|V’ (E restricted to V’) = { {u,v} ∈ E | {u,v} ⊆ V’ } • Example: • root : c • internal nodes : a,b,c • leaves: d,e,f,g • parent, child : (c,a),(a,b),… • sibling: (b,d), (a,e),(f,g) • ancestor, descendant: (c,b),(a,b),…

9. m-ary tree Def: A rooted tree is an m-ary tree (m > 0) if every internal vertex has no more than m children. • A tree is a full m-ary tree if every internal vertex has exactly m children. • A 2-ary tree is also called a binary tree. • Ordered rooted tree: the children of each vertex are ordered. • The first (second) child of a vertex in an ordered binary tree is called its left (right) child. • The subtree rooted at the left [right] child of the root of an ordered binary tree is called its left [right] subtree.

10. a c b g d f e h i k j l m Example 4: • left and right child of d = ? • left and right subtree of c = ? Notes: 1. Like graphs, there is no standard terminology used to describe trees, rooted trees , ordered rooted trees and binary trees. 2. Readers should carefully check meanings given to terms dealing with trees whenever they occur.

11. Trees as models • Molecule structure: • atom  vertex; bond  edge • e.g., saturated hydrocarbons CnH2n+2. • where C: degree = 4; H : degree = 1 • => there are (4n+2n+ 2)/2 = 3n+1 edges = #atoms -1 • => the connected graph must be a tree. • => non-isomorphic trees CnH2n+2 are isomers of CnH2n+2 • => for n = 4 => 2 isomers: butane(丁烷) : C-C-C-C • and isobutane(異丁烷) : • Representing organizations • Computer file system • tree-connected parallel processors C-C-C | C

12. Properties of trees Theorem 2: A tree with n ( n >0) vertices has exactly n-1 edges. Pf: Let T = (V,E) be a tree and choose a vertex r as the root. Now define the mapping g: EV- {r} by g(e) = v iff e={u,v} and v is the child of u. Obviously g is 1-1 and onto. Hence |E| = |V|-1. 1-1: if e1 = {u1,v}, e2 = {u2,v} with g(e1) = g(e2)=v => both u1 and u2 are the vertex appearing immediately before v in the unique simple path from r to v => u1 = u2. => if e1=e2 onto: v ∈ V – {r}, there is a unique simple path of length > 0 from r to v => there is a vertex u occurring immediately before v => g({u,v}) = v.

13. b d e h i Number of vertices in a complete m-ary tree Theorem 3: A full m-ary tree with i internal vertices contains n = mi + 1 vertices. Pf: Note: every edge in a tree T=(V,E,r) connects one parent(out-degree  ) and one child (in-degree). => the number of edges = |E| = Sv∈V #in-degree(v) [ = |V|-1 : another proof of theorem 2 ] = Sv∈V #out-degree(V) = Sv∈V #children(v) = Sv∈V-all leaves #children(v) --- leaves have no children. = Sv∈V-all leaves m = mi ----- every internal vertex has m children. => |V| = mi + 1.

14. Theorem 4 T: a full m-ary tree with n vertices, i internal vertices and l leaves. Then • 1. i = (n-1)/m and l = [(m-1)n+1]/m • 2. n = mi +1 and l = (m-1) x i + 1 • 3. n = (ml-1) / (m-1) and i = (l-1)/(m-1). pf: All are simple results of the following equations: (1) n = mi + 1 (*, Theorem 3: mi = |E| = |V|-1 =n-1) and (2) n = l + i (every vertex is either a leaf or an internal vertex) 1: i = (n-1) /m and l = n - i =n – (n-1)/m =[(m-1)n+1]/m 2,3: Left as exercises.

15. Example 9 Someone starts a chain letter. Each person receiving the letter either sends to 4 other people or stops sending letter. ==> • #people seeing the letter = ? if no one receiving the letter more than once and the chain letter ends after 100 people reading it but did not send it out. • #people sending out the letter = ? Sol: T=(V,E,r) where V = all people receiving (and seeing) the letter r = the person starting the game E = {(u,v) | u sends a letter to v } • #leaves = l = 100 => • n = l + #internal vertices(i) and n = 4i + 1 • n = l + (n-1) /4 => l = n – (n-1)/4 = (3n+1)/4 =100 • n = 400 –1 /3 = 133. => i = n –l = 33.

16. a c b g d f e h i k j l m the Level of nodes and the height of a tree T=(V,E,r) : a tree, a: a vertex. • Level(a) = depth(a) = the length of the unique path from root to a. • Level(T) = height(T)= max v∈ V level(v) • Example 10: • level(v) = ? for each vertex v in the tree? • Height(T) = ? • A m-ary tree is a complete tree if it is full and all leaves have the same height

17. Balanced trees • A tree isbalanced if all leaves have approximately the same depth. Def: A rooted m-ary tree of height h is called balanced if all leaves are at levels h or h-1. Example 11: Which of the rooted trees are balanced ? T2 T1

18. T3

19. Maximum number of leaves in a tree. Theorem 5: There are at most mh leaves in an m-ary tree T of height h. Pf: By math ind on h. Basis: h = 0 => T has only 1 = m0 node, which is a leaf. Ind. case: Assume every m-ary tree of height h has leaves ≤ mh => m-ary tree T of height h+1 has at most m subtrees, each of height h => #leaves(T) = St’ is a subtree of T #leaves(T’) ≤ m x mh = m h+1. Theorem 5’ : Every complete m-ary tree of height h has mh leaves. Pf: Like Theorem 5.

20. Corollary 1 • Corollary 1. 1. If an m-ary tree of height h has l leaves, then h ≥ [logm l]. 2. If the m-ary tree is full and balanced, then h =[logm l], where […] is the ceiling function. Pf:1. By Theorem 5, l ≤ m h; hence h ≥ log m l. Since h is an integer, h ≥ [log m l]. 2. ∵ balanced => h-1 ≤ level of leaves ≤ h ∵ height = h => Some vertex is at level h. => mh-1 < l ≤ m h. => h-1< log m l ≤ h => log m l ≤ h < 1 + log m l => h = [log m l].