Asymptotic estimate to derive an asymptotic formula

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Asymptotic estimate To derive an asymptotic formula of s n , we develop the generating function around its smallest singularity (Flajolet & Odlyzko, 1990), i.e. the radius of convergence of the power series. Since 1 - 6 z + z 2 = 1 - (3 - 8) z 1 - (3 + 8) z The smallest singular value is r 1 = 1 (3 + 8) and the asymptotic formula will have the exponential term r - n 1 = (3 + 8) n = (1 + 2) 2 n 17
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In a neighborhood of r 1 , the generating function can be rewritten as S ( z ) (1 + 2) 1 - 2 1 / 4 q 1 - (3 + 8) z + O (1 - (3 + 8) z ) 3 / 2 Since [ z n ] 1 - az ≈ - a n 4 πn 3 where [ z n ] F ( z ) denotes the n-th coefficient in the formal series of F, we have s n (1 + 2)(3 + 8) n 2 3 / 4 πn 3 = (1 + 2) 2 n +1 2 3 / 4 πn 3 Comparing with the number of binary trees, we have s n 1 . 44(1 . 46) n b n B.3 U NARY - BINARY EXPRESSIONS In the binary case, the number of expressions can be derived from the number of trees. This cannot be done in the unary-binary case, as the number of leaves in a tree with n internal nodes depends on the number of binary operators ( n 2 + 1 ). Generating function The number of trees with n internal nodes and n 2 binary operators can be derived from the following observation: any unary-binary tree with n 2 binary internal nodes can be generated from a binary tree by adding unary internal nodes. Each node in the binary tree can receive one or several unary parents. Since the binary tree has 2 n 2 + 1 nodes and the number of unary internal nodes to be added is n - n 2 , the number of unary-binary trees that can be created from a specific binary tree is the number of multisets with 2 n 2 + 1 elements on n - n 2 symbols, that is n + n 2 n - n 2 = n + n 2 2 n 2 If b q denotes the q-th Catalan number, the number of trees with n 2 binary operators among n is n + n 2 2 n 2 b n 2 Since such trees have n 2 + 1 leaves, with L leaves, p 2 binary and p 1 unary operators to choose from, the number of expressions is E ( n, n 2 ) = n + n 2 2 n 2 b n 2 p n 2 2 p n - n 2 1 L n 2 +1 Summing over all values of n 2 (from 0 to n ) yields the number of different expressions E n = n X n 2 =0 n + n 2 2 n 2 b n 2 p n 2 2 p n - n 2 1 L n 2 +1 z n Let E ( z ) be the corresponding generating function. E ( z ) = X n =0 E n z n = X n =0 n X n 2 =0 n + n 2 2 n 2 b n 2 p n 2 2 p n - n 2 1 L n 2 +1 z n = L X n =0 n X n 2 =0 n + n 2 2 n 2 b n 2 Lp 2 p 1 n 2 p n 1 z n = L X n =0 X n 2 =0 n + n 2 2 n 2 b n 2 Lp 2 p 1 n 2 ( p 1 z ) n 18
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since ( n + n 2 2 n 2 ) = 0 when n > n 2 E ( z ) = L X n 2 =0 b n 2 Lp 2 p 1 n 2 X n =0 n + n 2 2 n 2 ( p 1 z ) n = L X n 2 =0 b n 2 Lp 2 p 1 n 2 X n =0 n + 2 n 2 2 n 2 ( p 1 z ) n + n 2 = L X n 2 =0 b n 2 ( Lp 2 z ) n 2 X n =0 n + 2 n 2 2 n 2 ( p 1 z ) n applying the binomial formula E ( z ) = L X n 2 =0 b n 2 ( Lp 2 z ) n 2 1 (1 - p 1 z ) 2 n 2 +1 = L 1 - p 1 z X n 2 =0 b n 2 Lp 2 z (1 - p 1 z ) 2 n 2 applying the generating function for binary trees E ( z ) = L 1 - p 1 z 1 - q 1 - 4 Lp 2 z (1 - p 1 z ) 2 2 Lp 2 z (1 - p 1 z ) 2 = 1 - p 1 z 2 p 2 z 1 - s 1 - 4 Lp 2 z (1 - p 1 z ) 2 !
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