7-Trees and Stack Permutations

# 7-Trees and Stack Permutations - Trees Vertices(or nodes...

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Trees Vertices (or nodes) , edges, adjacent vertices, paths, simple paths, cycles, simple cycles of undirected graphs - no self references, no multiple edges. Connected graphs, components Let G be a graph with n>1 vertices.. Then G is a free tree iff 1. G is connected and has no simple cycles. 2. G has no simple cycles and n-1 edges. 3. G is connected and has n-1 edges. 4. G has no simple cycles and adding an edge between two nonadjacent vertices forms exactly one simple cycle. 5. G is connected and if an edge is deleted G becomes disconnected. 6. There is exactly one path between any two vertices. An oriented tree is a free tree with a designated vertex called the root . (Also called a rooted tree.) An ordered tree is an oriented tree with zero or more ordered subtrees T 1 , T 2 , . . ., T n , which are also ordered trees. Summary: Free tree - no root, no order Oriented tree - Free tree, root Ordered tree - Free tree, root, order. For n=4 nodes: 0 0 0 0 0 0 0 0 0 0 0 | / | \ / \ | | / | \ / \ / \ | | / | \ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | | / \ | | | / \ 0 0 0 0 0 0 0 0 0 0 | | | 0 0 0 2 Free 4 Oriented 5 Ordered For n=5 nodes: 0 0 0 0 0 0 0 0 0 0 0 0 | / \ / | | \ / \ / \ / \ | | | | / | \ / | | \ 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 | / \ | | | / \ | / \ / | \ | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | | | / \ 0 0 0 0 0 0 | | 0 0 3 Free 9 Oriented (14 Ordered not shown) Binary trees, complete binary trees, full binary trees.

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In a full binary tree the #leaf nodes = # internal nodes +1 Proof: Suppose # of nodes is n. Each node except the root has one edge coming into it and each internal node has exactly two edges coming out of it. The number of nodes is the # leaf nodes + number of internal nodes and the number of edges coming into all nodes is the same as the number coming out of all nodes. So, # edges coming in=n-1 # edges coming out =2(#internal nodes) n=#internal nodes + # leaf nodes # edgs coming in= # edges coming out Thus, n-1=2(#internal nodes)=#internal nodes + #leaf nodes - 1 Or, # leaf nodes=#internal nodes + 1 The number of binary trees with n nodes Construction: 0 / \ 0 < i < n i n-i Let b i be the number of binary trees with i nodes. Then b 0 =1 and b n = b 0 x b n-1 + b 1 x b n-2 + . . . + b n-1 x b 0 for n>0. So b 1 = b 0 x b 0 =1 b 2 = b 0 x b 1 + b 1 x b 0 = 1 x 1 + 1 x 1 =2 b 3 = b 0 x b 2 + b 1 x b 1 + b 2 x b 0 = 1 x 2 + 1 x 1 + 2 x 1 = 5 b 4 = b 0 x b 3 + b 1 x b 2 + b 2 x b 1 + b 3 x b 0 = 1 x 5 + 1 x 2 + 2 x 1 + 5 x 1 = 14 b 5 = . . Ordered forest - collection of zero or more ordered trees with each tree itself being given a designated order - 1 st , 2 nd , etc 1-1 correspondence n node Ordered forests n node Binary trees Give the construction! Indicate the recursive version of the transformation from an ordered forest of n nodes to its corresponding n node binary tree.
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