The average size of r-fold cut trees
\mathbb{E} X_{n,r} = \frac{1}{C_{n-1}} [z^n] \frac{u^{r+1}}{(1+u) (1 - u^{r+1})}
| >>> 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)) |
| >>> 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)) |
| >>> 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)) |
Theorem.
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}).