1 p 1 z p 1 p 1 z 2 4 lp 2 z 2 p 2 z reducing we have

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= 1 - p 1 z - p (1 - p 1 z ) 2 - 4 Lp 2 z 2 p 2 z Reducing, we have E ( z ) = 1 - p 1 z - p 1 - 2( p 1 + 2 Lp 2 k ) z + p 1 z 2 2 p 2 z Calculation As before, there is no closed simple formula for E n , but we can derive a recurrence formula by differentiating the generating function, rewritten as 2 p 2 zE ( z ) + p 1 z - 1 = - p 1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 2 p 2 zE 0 ( z ) + 2 p 2 E ( z ) + p 1 = p 1 + 2 p 2 L - p 1 z p 1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 2 p 2 zE 0 ( z ) + 2 p 2 E ( z ) + p 1 = ( p 1 + 2 p 2 L - p 1 z )(1 - p 1 z - 2 p 2 zE ( z )) 1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 2 p 2 zE 0 ( z ) + 2 p 2 E ( z ) 1 + z ( p 1 + 2 p 2 L - p 1 z ) 1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 = ( p 1 + 2 p 2 L - p 1 z )(1 - p 1 z ) 1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 - p 1 2 p 2 zE 0 ( z ) + 2 p 2 E ( z ) 1 - ( p 1 + 2 p 2 L ) z 1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 = 2 p 2 L (1 + p 1 z ) + p 1 ( p 1 - 1) z 1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 2 p 2 zE 0 ( z )(1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 ) + 2 p 2 E ( z )(1 - ( p 1 + 2 p 2 L ) z ) = (2 p 2 L (1 + p 1 z ) + p 1 ( p 1 - 1) z ) replacing E ( z ) and E 0 ( z ) with their coefficients 2 p 2 ( nE n - 2( p 1 + 2 p 2 L )( n - 1) E n - 1 + p 1 ( n - 2) E ( n - 2)) + 2 p 2 ( E n - ( p 1 + 2 p 2 L ) E n - 1 ) = 0 ( n + 1) E n - ( p 1 + 2 p 2 L )(2 n - 1) E n - 1 + p 1 ( n - 2) E n - 2 = 0 ( n + 1) E n = ( p 1 + 2 p 2 L )(2 n - 1) E n - 1 - p 1 ( n - 2) E n - 2 19
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which together with E 0 = L E 1 = ( p 1 + p 2 L ) L provides a formula for calculating E n . Asymptotic estimate As before, approximations of E n for large n can be found by developing E ( z ) in the neighbourhood of the root with the smallest module of 1 - 2( p 1 + 2 p 2 L ) z + p 1 z 2 The roots are r 1 = p 1 p 1 + 2 p 2 L - p p 2 1 + 4 p 2 2 L 2 + 4 p 2 p 1 L - p 1 r 2 = p 1 p 1 + 2 p 2 L + p p 2 1 + 4 p 2 2 L 2 + 4 p 2 p 1 L - p 1 both are positive and the smallest one is r 2 To alleviate notation, let δ = q p 2 1 + 4 p 2 2 L 2 + 4 p 2 p 1 L - p 1 r 2 = p 1 p 1 + 2 p 2 L + δ developing E ( z ) near r 2 , E ( z ) 1 - p 1 r 2 - q 1 - r 2 ( p 1 +2 p 2 L - δ p 1 ) q 1 - z r 2 2 p 2 r 2 + O (1 - z r 2 ) 3 / 2 E ( z ) p 1 + 2 p 2 L + δ - p 2 1 - p 1 + 2 p 2 L + δ 2 δ q 1 - z r 2 2 p 2 p 1 + O (1 - z r 2 ) 3 / 2 and therefore E n δr - n - 1 2 2 2 p 2 p 2 πp 1 n 3 = δ 2 p 2 2 πn 3 ( p 1 + 2 p 2 L + δ ) n + 1 2 p n +1 1 C G ENERATING RANDOM EXPRESSIONS In this section we present algorithms to generate random expressions with n internal nodes. We achieve this by generating random trees, and selecting randomly their nodes and leaves. We begin with the simpler binary case ( p 1 = 0 ). C.1 B INARY TREES To generate a random binary tree with n internal nodes, we use the following one-pass procedure. Starting with an empty root node, we determine at each step the position of the next internal nodes among the empty nodes, and repeat until all internal nodes are allocated. Start with an empty node, set e = 1 ; while n > 0 do Sample a position k from K ( e, n ) ; Sample the k next empty nodes as leaves; Sample an operator, create two empty children; Set e = e - k + 1 and n = n - 1 ; end Algorithm 1: Generate a random binary tree 20
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We denote by e the number of empty nodes, by n > 0 the number of operators yet to be generated, and by K ( e, n ) the probability distribution of the position ( 0 -indexed) of the next internal node to allocate.
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