On the number of subtrees on the fringe of random trees (partly - PowerPoint PPT Presentation

On the number of subtrees on the fringe of random trees (partly joined with Huilan Chang) Michael Fuchs Institute of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan April 16th, 2008 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 1 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 7 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 7 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 1 7 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 1 7 6 8 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 1 7 6 8 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 1 7 3 6 8 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 1 7 3 6 8 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 X 8 , 1 = 2 1 7 3 6 8 2 2 5 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 X 8 , 1 = 2 X 8 , 2 = 2 1 7 3 3 6 6 8 2 2 5 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 X 8 , 1 = 2 X 8 , 2 = 2 X 8 , 3 = 1 1 1 7 3 3 6 8 2 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 X 8 , 1 = 2 X 8 , 2 = 2 X 8 , 3 = 1 1 7 X 8 , 4 = 0 3 6 8 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 X 8 , 1 = 2 X 8 , 2 = 2 X 8 , 3 = 1 1 7 7 X 8 , 4 = 0 X 8 , 5 = 1 3 6 6 8 8 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 X 8 , 1 = 2 X 8 , 2 = 2 X 8 , 3 = 1 1 7 X 8 , 4 = 0 X 8 , 5 = 1 3 6 8 X 8 , 6 = 0 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 X 8 , 1 = 2 X 8 , 2 = 2 X 8 , 3 = 1 1 7 X 8 , 4 = 0 X 8 , 5 = 1 3 6 8 X 8 , 6 = 0 X 8 , 7 = 0 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

The number of subtrees X n,k = number of subtrees of size k on the fringe of random binary search trees of size n . Example : Input: 4 , 7 , 6 , 1 , 8 , 5 , 3 , 2 4 4 X 8 , 1 = 2 X 8 , 2 = 2 X 8 , 3 = 1 1 1 7 7 X 8 , 4 = 0 X 8 , 5 = 1 3 3 6 6 8 8 X 8 , 6 = 0 X 8 , 7 = 0 X 8 , 8 = 1 2 2 5 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 32

Mean value and variance X n,k satisfies d = X I n ,k + X ∗ X n,k n − 1 − I n ,k , where X k,k = 1 , X I n ,k and X ∗ n − 1 − I n ,k are conditionally independent given I n , and I n = Unif { 0 , . . . , n − 1 } . Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 3 / 32

Mean value and variance X n,k satisfies d = X I n ,k + X ∗ X n,k n − 1 − I n ,k , where X k,k = 1 , X I n ,k and X ∗ n − 1 − I n ,k are conditionally independent given I n , and I n = Unif { 0 , . . . , n − 1 } . This yields 2( n + 1) µ n,k := E ( X n,k ) = ( k + 1)( k + 2) , ( n > k ) , and 2 k (4 k 2 + 5 k − 3)( n + 1) σ 2 n,k := Var( X n,k ) = ( k + 1)( k + 2) 2 (2 k + 1)(2 k + 3) for n > 2 k + 1 . Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 3 / 32

Some previous results Aldous (1991): Weak law of large numbers X n,k − → 1 in probability . µ n,k Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 4 / 32

Some previous results Aldous (1991): Weak law of large numbers X n,k − → 1 in probability . µ n,k Devroye (1991): Central limit theorem X n,k − µ n,k d − → N (0 , 1) . σ n,k Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 4 / 32

Some previous results Aldous (1991): Weak law of large numbers X n,k − → 1 in probability . µ n,k Devroye (1991): Central limit theorem X n,k − µ n,k d − → N (0 , 1) . σ n,k Flajolet, Gourdon, Martinez (1997): Central limit theorem with optimal Berry-Esseen bound and LLT Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 4 / 32

Some previous results Aldous (1991): Weak law of large numbers X n,k − → 1 in probability . µ n,k Devroye (1991): Central limit theorem X n,k − µ n,k d − → N (0 , 1) . σ n,k Flajolet, Gourdon, Martinez (1997): Central limit theorem with optimal Berry-Esseen bound and LLT − → All the above results are for fixed k . Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 4 / 32

Results for k = k n Theorem (Feng, Mahmoud, Panholzer (2008)) (i) (Normal range) Let k = o ( √ n ) and k → ∞ as n → ∞ . Then, X n,k − µ n,k d − → N (0 , 1) . � 2 n/k 2 (ii) (Poisson range) Let k ∼ c √ n as n → ∞ . Then, d → Poisson(2 c − 2 ) . − X n,k (iii) (Degenerate range) Let k < n and √ n = o ( k ) as n → ∞ . Then, L 1 X n,k − → 0 . Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 5 / 32

Why are we interested in X n,k ? X n,k is a new kind of profile of a tree. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 6 / 32

Why are we interested in X n,k ? X n,k is a new kind of profile of a tree. The phase change from normal to Poisson is a universal phenomenon expected to hold for many classes of random trees. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 6 / 32

Why are we interested in X n,k ? X n,k is a new kind of profile of a tree. The phase change from normal to Poisson is a universal phenomenon expected to hold for many classes of random trees. The methods for proving phase change results might be applicable to other parameters which are expected to exhibit the same phase change behavior as well. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 6 / 32

Why are we interested in X n,k ? X n,k is a new kind of profile of a tree. The phase change from normal to Poisson is a universal phenomenon expected to hold for many classes of random trees. The methods for proving phase change results might be applicable to other parameters which are expected to exhibit the same phase change behavior as well. X n,k is related to parameters arising in genetics. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 6 / 32

Yule generated random genealogical trees Example: 5 4 time 3 2 1 genes 1 2 3 4 5 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 32

Yule generated random genealogical trees Example: Random model: At every time point, two yellow nodes uniformly coalescent. 1 2 3 4 5 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 32