On the number of subtrees on the fringe of random trees (partly joint with Huilan Chang)

On the number of subtrees on the fringe of random trees (partly joint 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 / 24

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 / 24

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 / 24

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 / 24

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 / 24

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 / 24

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 / 24

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 / 24

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 / 24

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 / 24

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 = 3 1 7 3 6 8 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 24

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 = 3 X 8,2 = 2 1 7 3 6 8 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 24

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 X 8,1 = 3 X 8,2 = 2 X 8,3 = 1 3 6 8 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 24

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 X 8,1 = 3 X 8,2 = 2 X 8,3 = 1 X 8,4 = 1 3 6 8 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 24

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 X 8,1 = 3 X 8,2 = 2 X 8,3 = 1 X 8,4 = 1 X 8,5 = 0 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 24

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 X 8,1 = 3 X 8,2 = 2 X 8,3 = 1 X 8,4 = 1 X 8,5 = 0 X 8,6 = 0 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 24

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 X 8,1 = 3 X 8,2 = 2 X 8,3 = 1 X 8,4 = 1 X 8,5 = 0 X 8,6 = 0 X 8,7 = 0 2 5 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 24

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 X 8,1 = 3 X 8,2 = 2 X 8,3 = 1 X 8,4 = 1 X 8,5 = 0 X 8,6 = 0 X 8,7 = 0 X 8,8 = 1 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 2 / 24

Mean value and variance X n,k satisfies d X n,k = XIn,k + Xn 1 I, n,k where X k,k = 1, X In,k and Xn 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 / 24

Mean value and variance X n,k satisfies X n,k d = XIn,k + X n 1 I n,k, where X k,k = 1, X In,k and Xn 1 I n,k are conditionally independent given I n, and I n = Unif{0,..., n 1}. This yields and for n > 2k + 1. µ n,k := E(X n,k ) = σ 2 n,k := Var(X n,k) = 2(n + 1), (n > k), (k + 1)(k + 2) 2k(4k 2 + 5k 3)(n + 1) (k + 1)(k + 2) 2 (2k + 1)(2k + 3) Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 3 / 24

Some previous results Aldous (1991): Weak law of large numbers X n,k µ n,k 1 in probability. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 4 / 24

Some previous results Aldous (1991): Weak law of large numbers X n,k µ n,k 1 in probability. Devroye (1991): Central limit theorem X n,k µ n,k σ n,k d N (0, 1). Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 4 / 24

Some previous results Aldous (1991): Weak law of large numbers X n,k µ n,k 1 in probability. Devroye (1991): Central limit theorem X n,k µ n,k σ n,k Flajolet, Gourdon, Martinez (1997): d N (0, 1). Central limit theorem with optimal Berry-Esseen bound and LLT Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 4 / 24

Some previous results Aldous (1991): Weak law of large numbers X n,k µ n,k 1 in probability. Devroye (1991): Central limit theorem X n,k µ n,k σ n,k Flajolet, Gourdon, Martinez (1997): d N (0, 1). 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 / 24

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 2n/k 2 d N (0, 1). (ii) (Poisson range) Let k c n as n. Then, X n,k d Poisson(2c 2 ). (iii) (Degenerate range) Let k < n and n = o(k) as n. Then, L X 1 n,k 0. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 5 / 24

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 / 24

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 / 24

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 / 24

Yule generated random genealogical trees Example: 5 4 3 2 time 1 1 2 3 4 5 6 genes Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 24

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 / 24

Yule generated random genealogical trees Example: Random model: At every time point, two yellow nodes uniformly coalescent. 1 1 2 3 4 5 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 24

Yule generated random genealogical trees Example: Random model: 2 At every time point, two yellow nodes uniformly coalescent. 1 1 2 3 4 5 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 24

Yule generated random genealogical trees Example: Random model: 3 2 At every time point, two yellow nodes uniformly coalescent. 1 1 2 3 4 5 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 24

Yule generated random genealogical trees Example: Random model: 3 4 2 At every time point, two yellow nodes uniformly coalescent. 1 1 2 3 4 5 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 24

Yule generated random genealogical trees Example: 5 Random model: 3 4 2 At every time point, two yellow nodes uniformly coalescent. 1 1 2 3 4 5 6 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 24

Yule generated random genealogical trees Example: 5 Random model: 3 4 2 At every time point, two yellow nodes uniformly coalescent. 1 1 2 3 4 5 6 Same model as random binary search tree model! Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 7 / 24

Shape parameters of genealogical trees k-pronged nodes (Rosenberg 2006): Nodes with an induced subtree with k 1 internal nodes. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 8 / 24

Shape parameters of genealogical trees k-pronged nodes (Rosenberg 2006): Nodes with an induced subtree with k 1 internal nodes. k-caterpillars (Rosenberg 2006): Correspond to nodes whose induced subtree has k 1 internal nodes all of them with out-degree either 0 or 1. Nodes with minimal clade size k (Blum and François (2005)): If k 3, then they are internal nodes with induced subtree of size k 1 and either an empty right subtree or empty left subtree. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 8 / 24

Counting pattern in random binary search trees Consider X n,k with d X n,k = XIn,k + Xn 1 I, n,k where X k,k = Bernoulli(p k ), X In,k and Xn 1 I n,k are conditionally independent given I n, and I n = Unif{0,..., n 1}. Then, p k shape parameter 1 # of k + 1-pronged nodes 2/k # of nodes with minimal clade size k + 1 2 k 1 /k! # of k + 1 caterpillars Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 9 / 24

Underlying recurrence and solution All (centered or non-centered) moments satisfy a n,k = 2 n 1 a j,k + b n,k, n j=0 where a k,k is given and a n,k = 0 for n < k. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 10 / 24

Underlying recurrence and solution All (centered or non-centered) moments satisfy a n,k = 2 n 1 a j,k + b n,k, n j=0 where a k,k is given and a n,k = 0 for n < k. We have a n,k = where n > k. 2(n + 1) (k + 1)(k + 2) a k,k + 2(n + 1) k<j<n b j,k (j + 1)(j + 2) + b n,k, Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 10 / 24

Mean value and variance We have and E(X n,k ) = 2(n + 1) (k + 1)(k + 2) p k, (n > k), Var(X n,k ) = 2p k(4k 3 + 16k 2 + 19k + 6 (11k 2 + 22k + 6)p k )(n + 1) (k + 1)(k + 2) 2 (2k + 1)(2k + 3) for n > 2k + 1. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 11 / 24

Mean value and variance We have and E(X n,k ) = 2(n + 1) (k + 1)(k + 2) p k, (n > k), Var(X n,k ) = 2p k(4k 3 + 16k 2 + 19k + 6 (11k 2 + 22k + 6)p k )(n + 1) (k + 1)(k + 2) 2 (2k + 1)(2k + 3) for n > 2k + 1. Note that E(X n,k ) Var(X n,k ) 2p k k 2 n for n > 2k + 1 and k as n. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 11 / 24

Higher moments Denote by A (m) n,k := E(X n,k E(X n,k )) m. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 12 / 24

Higher moments Denote by Then, where A (m) n,k := E(X n,k E(X n,k )) m. A (m) n,k = 2 n 1 n j=0 A (m) j,k + B(m) n,k, B (m) n,k := i 1 +i 2 +i 3 =m 0 i 1,i 2 <m ( m ) n 1 1 i 1, i 2, i 3 n j=0 A (i 1) j,k A(i 2) n 1 j,k i 3 n,j,k and n,j,k = E(X j,k ) + E(X n 1 j,k ) E(X n,k ). Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 12 / 24

Normal range Proposition Uniformly for n, k, m 1 and n > k ( A (m) n,k = O max { 2p k n k 2, ( ) }) 2pk n m/2. k 2 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 13 / 24

Normal range Proposition Uniformly for n, k, m 1 and n > k ( A (m) n,k = O max { 2p k n k 2, ( ) }) 2pk n m/2. k 2 Proposition For E(X n,k ) as n, ( (2pk ) ) n m 1/2 = o A (2m 1) n,k k 2, A (2m) n,k ( ) 2pk n m g m k 2, where g m = (2m)!/(2 m m!). Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 13 / 24

Poisson range Consider Ā (m) n,k = E(X n,k(x n,k 1) (X n,k m + 1)). Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 14 / 24

Poisson range Consider Ā (m) n,k = E(X n,k(x n,k 1) (X n,k m + 1)). Then, similarly as before: Proposition (i) Uniformly for n, k, m 1 and n > k { ( ) Ā (m) n,k (max = O 2pk n k 2, 2pk n m }) k 2. (ii) For E(X n,k ) c and k < n as n, Ā (m) n,k cm. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 14 / 24

The phase change Theorem (i) (Normal range) Let E(X n,k ) and k as n. Then, X n,k E(X n,k ) 2pk n/k 2 d N (0, 1). (ii) (Poisson range) Let E(X n,k ) c > 0 and k < n as n. Then, X n,k d Poisson(c). (iii) (Degenerate range) Let E(X n,k ) 0 as n. Then, L X 1 n,k 0. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 15 / 24

A comparison of the phase change For k-caterpillars, we have Note that either So, there is no Poisson range. E(X n,k ) = 2k 1 n (k + 2)!. E(X n,k ) or E(X n,k ) 0. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 16 / 24

A comparison of the phase change For k-caterpillars, we have Note that either So, there is no Poisson range. E(X n,k ) = 2k 1 n (k + 2)!. E(X n,k ) or E(X n,k ) 0. shape parameter location phase change k-pronged nodes n normal - poisson - degenerate 3 minimal clade size k n normal - poisson - degenerate k-caterpillars ln n/(ln ln n) normal - degenerate Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 16 / 24

Refined results (for # of subtrees) Define and ( ) φ n,k (y) = e σ2 n,k y2 /2 E e (X n,k µ n,k )y. n,k = dm φ n,k (y) dy m. y=0 φ (m) Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 17 / 24

Refined results (for # of subtrees) Define and ( ) φ n,k (y) = e σ2 n,k y2 /2 E e (X n,k µ n,k )y. n,k = dm φ n,k (y) dy m. y=0 φ (m) Proposition Uniformly for n, k 1 and m 0 { φ (m) n ( n ) } m/3 n,k m!am max k 2, k 2 for a suitable constant A. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 17 / 24

Berry-Esseen bound and LLT for the normal range Theorem (Rate of convergency) For 1 k = o( n) as n, ( ) sup P X n,k µ n,k < x Φ(x) σ n,k = O x R ( k n ). Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 18 / 24

Berry-Esseen bound and LLT for the normal range Theorem (Rate of convergency) For 1 k = o( n) as n, ( ) sup P X n,k µ n,k < x Φ(x) σ n,k = O x R ( k n ). Theorem (LLT) For 1 k = o( n) as n, ( P (X n,k = µ n,k + xσ n,k ) = e x2 /2 (1 + O (1 + x 3 ) k )), 2πσn,k n uniformly in x = o(n 1/6 /k 1/3 ). Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 18 / 24

LLT for the Poisson range Define and φ n,k (y) = e µ n,k(y 1) E ( y X n,k ). n,k = dm φn,k (y) dy m. y=1 φ (m) Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 19 / 24

LLT for the Poisson range Define and φ n,k (y) = e µ n,k(y 1) E ( y X n,k ). n,k = dm φn,k (y) dy m. y=1 φ (m) Proposition Uniformly for n > k and m 0 for a suitable constant A. (m) φ ( n,k n ) m/2 m!am k 3 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 19 / 24

Poisson approximation Theorem (LLT) For k < n and n, P (X n,k = l) = e µ (µ n,k n,k) l l! uniformly in l. ( n ) + O k 3 Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 20 / 24

Poisson approximation Theorem (LLT) For k < n and n, P (X n,k = l) = e µ (µ n,k n,k) l l! uniformly in l. ( n ) + O k 3 Theorem (Poisson approximation) Let k < n and k as n. Then, d T V (X n,k, Poisson(µ n,k )) 0. Remark: A rate can be given as well. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 20 / 24

Other types of random trees Random recursive trees Non-plane, labelled trees with every label sequence from the root to a leave increasing; random model is the uniform model. Methods works as well (with minor modifications) and similar results can be proved. Plane-oriented recursive trees (PORTs) Plane, labelled trees with every label sequence from the root to a leave increasing; random model is the uniform model. Method works as well, but details more involved. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 21 / 24

Mean value and variance of PORTs We have, µ n,k := E(X n,k ) = 2n 1 4k 2 1, (n > k). Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 22 / 24

Mean value and variance of PORTs We have, µ n,k := E(X n,k ) = 2n 1 4k 2 1, (n > k). Moreover, for fixed k as n, where Var(X n,k ) c k n, c k = 8k2 4k 8 ((2k 3)!!) 2 (4k 2 1) 2 ((k 1)!) 2 4 k 1 k(2k + 1), and, for k < n and k as n, E(X n,k ) Var(X n,k ) n 2k 2. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 22 / 24

The phase change Theorem (i) (Normal range) Let k = o ( n) and k as n. Then, X n,k µ n,k n/(2k 2 ) d N (0, 1). (ii) (Poisson range) Let k c n as n. Then, X n,k d Poisson((2c 2 ) 1 ). (iii) (Degenerate range) Let k < n and n = o(k) as n. Then, L X 1 n,k 0. Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 23 / 24

More results and future research Parameters of genealogical trees under different random models Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 24 / 24

More results and future research Parameters of genealogical trees under different random models Universality of the phase change for the number of subtrees Very simple classes of increasing trees and more general classes of increasing trees (polynomial varieties, mobile trees, etc.) Phase change results for the number of nodes with out-degree k Important in computer science. A phase change from normal to degenerate is expected (no Poisson range). Michael Fuchs (NCTU) Subtree Sizes of Random Trees April 16th, 2008 24 / 24