A CONNECTION d = # of nodes at k left branches away from the root in X n,k random BSTs of n − 1 keys by ( i ) the usual transformation from a multiway tree to a binary tree (first branch → left, sibling → right) and ( ii ) the bijection between binary increasing trees and binary search trees. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.11/64
A CONNECTION d = # of nodes at k left branches away from the root in X n,k random BSTs of n − 1 keys by ( i ) the usual transformation from a multiway tree to a binary tree (first branch → left, sibling → right) and ( ii ) the bijection between binary increasing trees and binary search trees. It suffices to look at BSTs. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.11/64
A CONNECTION d = # of nodes at k left branches away from the root in X n,k random BSTs of n − 1 keys by ( i ) the usual transformation from a multiway tree to a binary tree (first branch → left, sibling → right) and ( ii ) the bijection between binary increasing trees and binary search trees. It suffices to look at BSTs. But ( i ) it’s much simpler to start from the better structured recursive trees; ( ii ) behaviors not identical in all ranges. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.11/64
SUMMARY OF MAIN PHENOMENA k Write throughout α n,k = and lim n α n,k = α . log n Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.12/64
SUMMARY OF MAIN PHENOMENA k Write throughout α n,k = and lim n α n,k = α . log n ➠ E ( X n,k ) unimodal, but V ( X n,k ) bimodal Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.12/64
SUMMARY OF MAIN PHENOMENA k Write throughout α n,k = and lim n α n,k = α . log n ➠ E ( X n,k ) unimodal, but V ( X n,k ) bimodal X n,k d ➠ If 0 ≤ α < e , then − → X α . E ( X n,k ) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.12/64
SUMMARY OF MAIN PHENOMENA k Write throughout α n,k = and lim n α n,k = α . log n ➠ E ( X n,k ) unimodal, but V ( X n,k ) bimodal X n,k d ➠ If 0 ≤ α < e , then − → X α . E ( X n,k ) X n,k m ➠ If 0 ≤ α ≤ 1 , then − → X α . E ( X n,k ) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.12/64
SUMMARY OF MAIN PHENOMENA ➠ For k = o ( log n ) , X n,k − E ( X n,k ) m − → N ( 0, 1 ) . � V ( X n,k ) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.13/64
SUMMARY OF MAIN PHENOMENA ➠ For k = o ( log n ) , X n,k − E ( X n,k ) m − → N ( 0, 1 ) . � V ( X n,k ) ➠ If k = log n + o ( log n ) and | k − log n | → ∞ , then X n,k − E ( X n,k ) m → X ′ − 1 . � V ( X n,k ) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.13/64
SUMMARY OF MAIN PHENOMENA ➠ For k = o ( log n ) , X n,k − E ( X n,k ) m − → N ( 0, 1 ) . � V ( X n,k ) ➠ If k = log n + o ( log n ) and | k − log n | → ∞ , then X n,k − E ( X n,k ) m → X ′ − 1 . � V ( X n,k ) ➠ For k = log n + O ( 1 ) , the limit law of X n,k − E ( X n,k ) does not exist. � V ( X n,k ) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.13/64
RECURRENCE OF X n,k Y d = X uniform [ 1,n − 1 ] ,k − 1 + X ∗ X n,k n − uniform [ 1,n − 1 ] ,k Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.14/64
RECURRENCE OF X n,k Y d = X uniform [ 1,n − 1 ] ,k − 1 + X ∗ X n,k n − uniform [ 1,n − 1 ] ,k Three proofs: ( i ) bijection conditioned on the size of the subtree rooted at 2; ( ii ) above-mentioned transformation; ( iii ) algebraic � n − 1 � � � ( j − 1 )! · · · ( j s − 1 )! 1 d = X n,k s ! j 1 , . . . , j s ( n − 1 )! � �� � s ≥ 1 j 1 + ··· + j s = n − 1 P ( s subtrees have sizes j 1 ,...,j s ) × ( X j 1 ,k − 1 + · · · + X j s ,k − 1 ) . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.14/64
EXPECTED VALUE OF X n,k Meir, Moon (1978) (implicit): for 0 ≤ k < n Stirling1 ( n, k + 1 ) µ n,k := E ( X n,k ) = ; ( n − 1 )! also in Moon (1974) and Dondajewski, Szyma´ nski (1982). Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.15/64
EXPECTED VALUE OF X n,k Meir, Moon (1978) (implicit): for 0 ≤ k < n Stirling1 ( n, k + 1 ) µ n,k := E ( X n,k ) = ; ( n − 1 )! also in Moon (1974) and Dondajewski, Szyma´ nski (1982). By known estimates for Stirling first numbers (H. 1995) ( log n ) k � � �� 1 µ n,k = 1 + O , k log n Γ ( 1 + log n ) k ! uniformly for 0 ≤ k ≤ K log n . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.15/64
A ROUGH DESCRIPTION OF SHAPE The root has about log n subtrees, Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.16/64
A ROUGH DESCRIPTION OF SHAPE The root has about log n subtrees, each of them “attracting” about the same number of new keys. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.16/64
A ROUGH DESCRIPTION OF SHAPE The root has about log n subtrees, each of them “attracting” about the same number of new keys. Also log µ n,k → α ( 1 − log α ) for 0 ≤ α ≤ K , log n Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.16/64
A ROUGH DESCRIPTION OF SHAPE The root has about log n subtrees, each of them “attracting” about the same number of new keys. Also log µ n,k → α ( 1 − log α ) for 0 ≤ α ≤ K , and log n µ n,k → ∞ when 1 ≤ k ≤ e log n − 1 2 log log n + O ( 1 ) . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.16/64
TWO COROLLARIES The estimate for µ n,k also implies ☞ an LLT for the depth; see Devroye (1988), Szyma´ nski (1990), Mahmoud (1991) for CLT, and Dobrow, Smythe (1996) for Poisson approximation; Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.17/64
TWO COROLLARIES The estimate for µ n,k also implies ☞ an LLT for the depth; see Devroye (1988), Szyma´ nski (1990), Mahmoud (1991) for CLT, and Dobrow, Smythe (1996) for Poisson approximation; ☞ that the expected height is bounded above by E ( H n ) ≤ e log n − 1 2 log log n + O ( 1 ) . (Roughly, the range when µ n,k → ∞ .) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.17/64
SECOND MOMENT OF X n,k Meir, Moon (1978) (implicit) � Stirling1 ( n, k + j + 1 ) � 2j � E ( X 2 n,k ) = ; j ( n − 1 )! 0 ≤ j ≤ k see also van der Hofstad et al. (2002); Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.18/64
SECOND MOMENT OF X n,k Meir, Moon (1978) (implicit) � Stirling1 ( n, k + j + 1 ) � 2j � E ( X 2 n,k ) = ; j ( n − 1 )! 0 ≤ j ≤ k see also van der Hofstad et al. (2002); and for k = O ( 1 ) ( log n ) 2k − 1 V ( X n,k ) ∼ ( 2k − 1 )( k − 1 )! 2 . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.18/64
VARIANCE OF X n,k : MIDDLE RANGE Uniformly for 1 ≤ k ≤ 2 log n − K √ log n V ( X n,k ) ∼ φ ( α n,k ) µ 2 n,k , where Γ ( x + 1 ) 2 φ ( x ) := 2 ) Γ ( 2x + 1 ) − 1. ( 1 − x A full asymptotic expansion can be derived. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.19/64
φ ( 0 ) = φ ( 1 ) = φ ′ ( 1 ) = 0 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.20/64
MORE PRECISE ESTIMATES ( log n ) 2k − 1 ( 2k − 1 )( k − 1 )! 2 , if k = o ( log n ) , V ( X n,k ) ∼ � 2 � ( log n ) k − 1 if t n := k − log n p ( t n ) , k ! = o ( log n ) , where � � � � 2 − π 2 2ζ ( 3 ) + 4γ + π 2 t 2 p ( t n ) := n − 3 ( 1 − γ ) − 6 t n + 6 2γ 2 − 6γ + 8 − 2ζ ( 3 )( 1 − γ ) − π 2 γ 2 − 2γ + 3 − π 4 � � 360 . 6 Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.21/64
MORE PRECISE ESTIMATES ( log n ) 2k − 1 ( 2k − 1 )( k − 1 )! 2 , if k = o ( log n ) , V ( X n,k ) ∼ � 2 � ( log n ) k − 1 if t n := k − log n p ( t n ) , k ! = o ( log n ) , where � � � � 2 − π 2 2ζ ( 3 ) + 4γ + π 2 t 2 p ( t n ) := n − 3 ( 1 − γ ) − 6 t n + 6 2γ 2 − 6γ + 8 − 2ζ ( 3 )( 1 − γ ) − π 2 γ 2 − 2γ + 3 − π 4 � � 360 . 6 φ ′′ ( 1 ) = 2 − π 2 2 6 Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.21/64
BIMODALITY OF VARIANCE WHEN α = 1 n 2 k = log n + O ( √ log n ) V ( X n,k ) ≍ ( log n ) 3 , min Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.22/64
BIMODALITY OF VARIANCE WHEN α = 1 n 2 k = log n + O ( √ log n ) V ( X n,k ) ≍ ( log n ) 3 , min 1 ≤ k<n V ( X n,k ) ∼ 12 − π 2 n 2 · max ( log n ) 2 . 12πe Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.22/64
BIMODALITY OF VARIANCE WHEN α = 1 n 2 k = log n + O ( √ log n ) V ( X n,k ) ≍ ( log n ) 3 , min 1 ≤ k<n V ( X n,k ) ∼ 12 − π 2 n 2 · max ( log n ) 2 . 12πe p ( t n ) ( log n ) 2 · µ 2 Note that V ( X n,k ) ∼ n,k . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.22/64
BIMODALITY OF VARIANCE WHEN α = 1 n 2 k = log n + O ( √ log n ) V ( X n,k ) ≍ ( log n ) 3 , min 1 ≤ k<n V ( X n,k ) ∼ 12 − π 2 n 2 · max ( log n ) 2 . 12πe p ( t n ) ( log n ) 2 · µ 2 Note that V ( X n,k ) ∼ n,k . More precise estimates can be derived. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.22/64
E ( X 500,k ) AND V ( X 500,k ) 1000 800 600 400 200 0 2 4 6 8 10 12 14 Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.23/64
A GLOBAL DESCRIPTION OF V ( X n,k ) 2α ( 1 − log α ) , if 0 ≤ α ≤ 2 ; log V ( X n,k ) 4 − α log 4, if 2 ≤ α ≤ 4 ; → log n if 4 ≤ α ≤ K. α ( 1 − log α ) , Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.24/64
A GLOBAL DESCRIPTION OF V ( X n,k ) 2α ( 1 − log α ) , if 0 ≤ α ≤ 2 ; log V ( X n,k ) 4 − α log 4, if 2 ≤ α ≤ 4 ; → log n if 4 ≤ α ≤ K. α ( 1 − log α ) , ≍ µ 2 if 0 ≤ α < 2 ; n,k , ≫ µ 2 Thus V ( X n,k ) if 2 ≤ α ≤ 4 ; n,k , µ n,k , ≍ µ n,k , if 4 < α ≤ K. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.24/64
LIMIT DISTRIBUTION: GENESIS Let ¯ X n,k := X n,k /µ n,k and I n := uniform [ 1, n − 1 ] . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.25/64
LIMIT DISTRIBUTION: GENESIS Let ¯ X n,k := X n,k /µ n,k and I n := uniform [ 1, n − 1 ] . Then from d = X I n ,k − 1 + X ∗ X n,k n − I n ,k , it follows that = µ I n ,k − 1 X I n ,k − 1 + µ n − I n ,k d ¯ ¯ ¯ X ∗ X n,k n − I n ,k . µ n,k µ n,k Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.25/64
LIMIT DISTRIBUTION: GENESIS Since µ n,k ≈ ( log n ) k /k ! and I n = ⌈ ( n − 1 ) U ⌉ , we expect that � k − 1 � log n + log U µ I n ,k − 1 k d → αU α , ≈ − µ n,k log n log n Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.26/64
LIMIT DISTRIBUTION: GENESIS Since µ n,k ≈ ( log n ) k /k ! and I n = ⌈ ( n − 1 ) U ⌉ , we expect that � k − 1 � log n + log U µ I n ,k − 1 k d → αU α , ≈ − µ n,k log n log n and similarly µ n − I n ,k d → ( 1 − U ) α . − µ n,k Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.26/64
LIMIT DISTRIBUTION: GENESIS Since µ n,k ≈ ( log n ) k /k ! and I n = ⌈ ( n − 1 ) U ⌉ , we expect that � k − 1 � log n + log U µ I n ,k − 1 k d → αU α , ≈ − µ n,k log n log n and similarly µ n − I n ,k d → ( 1 − U ) α . Thus if − µ n,k d d ¯ = αU α X α + ( 1 − U ) α X ∗ − X n,k → X α , then X α α . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.26/64
LIMIT DISTRIBUTIONS: NEW RESULTS For 0 ≤ α < e X n,k d − → X α , µ n,k with convergence of the first m moments for 0 ≤ α < m 1/ ( m − 1 ) . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.27/64
LIMIT DISTRIBUTIONS: NEW RESULTS For 0 ≤ α < e X n,k d − → X α , µ n,k with convergence of the first m moments for 0 ≤ α < m 1/ ( m − 1 ) . In particular, convergence of all moments holds only for α ∈ [ 0, 1 ] . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.27/64
LIMIT DISTRIBUTIONS: NEW RESULTS For 0 ≤ α < e X n,k d − → X α , µ n,k with convergence of the first m moments for 0 ≤ α < m 1/ ( m − 1 ) . In particular, convergence of all moments holds only for α ∈ [ 0, 1 ] . X 0 = X 1 ≡ 1 Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.27/64
RECURRENCE OF MOMENTS Let ν m := E ( X m α ) . Then ν 0 = ν 1 = 1 and � � m � 1 ν j ν m − j α j − 1 ν m = j m − α m − 1 1 ≤ j<m × Γ ( jα + 1 ) Γ (( m − j ) α + 1 ) ( m ≥ 2 ) , Γ ( mα + 1 ) for 0 ≤ α < m 1/ ( m − 1 ) . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.28/64
ASYMPTOTIC NORMALITY WHEN α = 0 If 1 ≤ k = o ( log n ) , then � � �� X n,k − ( log n ) k � � � k � � k ! − Φ ( x ) = O sup < x , P � � � log n � � ( log n ) 2k − 1 x � � ( k − 1 )! 2 ( 2k − 1 ) � � where Φ ( x ) denotes the standard normal distribution. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.29/64
ASYMPTOTIC NORMALITY WHEN α = 0 If 1 ≤ k = o ( log n ) , then � � �� X n,k − ( log n ) k � � � k � � k ! − Φ ( x ) = O sup < x , P � � � log n � � ( log n ) 2k − 1 x � � ( k − 1 )! 2 ( 2k − 1 ) � � where Φ ( x ) denotes the standard normal distribution. � X n,1 ∼ N ( log n, log n ) In particular, . . . X n,2 ∼ N ( 1 2 ( log n ) 2 , 1 3 ( log n ) 3 ) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.29/64
A QUICKSORT-TYPE LIMIT LAW WHEN α = 1 If t n := k − log n = o ( log n ) and t n → ∞ , then X n,k − µ n,k m → X ′ − 1 , ( log n ) k − 1 t n k ! where X ′ 1 = ( d X α / d α ) | α = 1 or ∗ + U + U log U + ( 1 − U ) log ( 1 − U ) . d X ′ = UX ′ 1 + ( 1 − U ) X ′ 1 1 Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.30/64
A QUICKSORT-TYPE LIMIT LAW WHEN α = 1 If t n := k − log n = o ( log n ) and t n → ∞ , then X n,k − µ n,k m → X ′ − 1 , ( log n ) k − 1 t n k ! where X ′ 1 = ( d X α / d α ) | α = 1 or ∗ + U + U log U + ( 1 − U ) log ( 1 − U ) . d X ′ = UX ′ 1 + ( 1 − U ) X ′ 1 1 Same law as total path length (or left path length in BST; Dobrow, Fill, 1999) and cost of an in-situ permutation algorithm. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.30/64
NONEXISTENCE OF LIMIT LAW WHEN k = log n + O ( 1 ) If k = log n + O ( 1 ) , then the limit distribution of � ( X n,k − µ n,k ) / V ( X n,k ) does not exist. Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.31/64
NONEXISTENCE OF LIMIT LAW WHEN k = log n + O ( 1 ) If k = log n + O ( 1 ) , then the limit distribution of � ( X n,k − µ n,k ) / V ( X n,k ) does not exist. Main step of the proof: � m � ( log n ) k − 1 E ( X n,k − µ n,k ) m ∼ Polynomial ( t n ) ; � �� � k ! degree = m the remaining proof is similar to that used for the space requirement of random m -ary search trees when m ≥ 27 . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.31/64
APPROACHES USED Convergence in distribution: contraction method Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.32/64
APPROACHES USED Convergence in distribution: contraction method Convergence of moments: method of moments Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.32/64
APPROACHES USED Convergence in distribution: contraction method Convergence of moments: method of moments Common to both approaches is the resolution of the double-indexed recurrence � 1 a n,k = b n,k + ( a j,k − 1 + a j,k ) . n − 1 1 ≤ j<n Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.32/64
APPROACHES USED Convergence in distribution: contraction method Convergence of moments: method of moments Common to both approaches is the resolution of the double-indexed recurrence � 1 a n,k = b n,k + ( a j,k − 1 + a j,k ) . n − 1 1 ≤ j<n (Martingale arguments also apply to recursive trees.) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.32/64
SOLUTION OF THE RECURRENCE If � 1 a n,k = ( a j,k + a j,k − 1 ) + b n,k , ( n ≥ 2 ; k ≥ 1 ) , n − 1 1 ≤ j<n with a n,k = b n,k for n = 1 and k = 0 , Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.33/64
SOLUTION OF THE RECURRENCE If � 1 a n,k = ( a j,k + a j,k − 1 ) + b n,k , ( n ≥ 2 ; k ≥ 1 ) , n − 1 1 ≤ j<n with a n,k = b n,k for n = 1 and k = 0 , then � j − 1 � � 1 + w � � b j,k − r [ w r ]( w + 1 ) a n,k = b n,k + . ℓ 2 ≤ j<n 0 ≤ r ≤ k j<ℓ<n Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.33/64
SOLUTION OF THE RECURRENCE If � 1 a n,k = ( a j,k + a j,k − 1 ) + b n,k , ( n ≥ 2 ; k ≥ 1 ) , n − 1 1 ≤ j<n with a n,k = b n,k for n = 1 and k = 0 , then � j − 1 � � 1 + w � � b j,k − r [ w r ]( w + 1 ) a n,k = b n,k + . ℓ 2 ≤ j<n 0 ≤ r ≤ k j<ℓ<n Proof by GF Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.33/64
WHICHEVER APPROACH REQUIRES MEAN VALUE µ n,k satisfies the recurrence with b n,k = δ 0,k . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.34/64
WHICHEVER APPROACH REQUIRES MEAN VALUE µ n,k satisfies the recurrence with b n,k = δ 0,k . Thus � � 1 1 + w � � µ n,k = [ w k ]( w + 1 ) j ℓ 2 ≤ j<n j<ℓ<n Stirling1 ( n, k + 1 ) = ( n − 1 )! ( log n ) k � � �� 1 = 1 + O . k log n Γ ( 1 + log n ) k ! Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.34/64
WHICHEVER APPROACH REQUIRES MEAN VALUE µ n,k satisfies the recurrence with b n,k = δ 0,k . Thus � � 1 1 + w � � µ n,k = [ w k ]( w + 1 ) j ℓ 2 ≤ j<n j<ℓ<n Stirling1 ( n, k + 1 ) = ( n − 1 )! ( log n ) k � � �� 1 = 1 + O . k log n Γ ( 1 + log n ) k ! Sufficient for all ranges except k = log n + O ( 1 ) . Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.34/64
ASYMPTOTICS OF HIGHER MOMENTS For method of moments: all moments (centered or not) satisfy the same type of recurrences � 1 a n,k = ( a j,k + a j,k − 1 ) + b n,k , n − 1 1 ≤ j<n with different b n,k , and we need the ∼ -transfer : Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.35/64
ASYMPTOTICS OF HIGHER MOMENTS For method of moments: all moments (centered or not) satisfy the same type of recurrences � 1 a n,k = ( a j,k + a j,k − 1 ) + b n,k , n − 1 1 ≤ j<n with different b n,k , and we need the ∼ -transfer : � m � m � � ( log n ) k ( log n ) k , then a n,k ∼ c mα + 1 if b n,k ∼ c . mα − α m Γ ( 1 + α ) k ! Γ ( 1 + α ) k ! c = c ( α ) Profile of random recursive trees and random binary search trees, INRIA, 26/04/2004 – p.35/64
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