$[z^n w^k] T(z,w)$ counts # of trees with $n$ inner nodes and $k$ leaves.
sage: A.<n> = AsymptoticRing('n^QQ', SR, default_prec=4)
sage: (1 + 1/n)^n
e - 1/2*e*n^(-1) + 11/24*e*n^(-2) - 7/16*e*n^(-3) + O(n^(-4))
>>> catalan_shifted_asy = asymptotic_expansions.Binomial_kn_over_n(n, k=2, precision=3)
>>> catalan_shifted_asy = catalan_shifted_asy.subs(n=n-1) / n
>>> catalan_shifted_asy
1/4/sqrt(pi)*4^n*n^(-3/2) + 3/32/sqrt(pi)*4^n*n^(-5/2) + 25/512/sqrt(pi)*4^n*n^(-7/2) + O(4^n*n^(-9/2))
>>> z_asy = 1/4 * (1 - Z^-1)
>>> u_asy = upsilon(z_asy) + O(Z^-2); u_asy
1 - 2*Z^(-1/2) + 2*Z^(-1) - 2*Z^(-3/2) + 2*Z^(-2) - 2*Z^(-5/2) + O(Z^(-2))
>>> singular_expansion = fast_callable(u^(r+1) / (1 + u) / (1 - u^(r+1)), vars=(u, r))(u_asy, r)
>>> singular_expansion
(1/4/(r + 1))*Z^(1/2) + 1/4*((r + 1)^2 + r + 1)/(r + 1)^2 - 1/2 + (1/2*r - 1/2*((r + 1)^2 + r + 1)/(r + 1) + 1/12*(3*((r + 1)^2 + r + 1)^2/(r + 1)^2 - (2*(r + 1)^3 + 3*(r + 1)^2 + 4*r + 4)/(r + 1))/(r + 1) + 1/2)*Z^(-1/2) + O(Z^(-1))
>>> expectation = singular_expansion._singularity_analysis_(n, zeta=1/4, precision=3) / catalan_shifted_asy
>>> expectation.map_coefficients(lambda t: t.simplify_rational())
(1/(r + 1))*n - 1/6*(r^2 - r)/(r + 1) + O(n^(-1/2))
After $r$ cuts, the average number of remaining nodes is \[ \mathbb{E} X_{n,r} = \frac{n}{r+1} - \frac{r(r-1)}{6(r+1)} + O(n^{-1/2}). \]
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