H - Beautiful Binary Tree Editorial by en_translator

It can be proved that a binary tree with \(0\) or \(1\) written on each vertex is a beautiful binary tree of degree \(N\) if and only if

  • the root and every leaf has \(1\) written on it;
  • the sum of integers written on all vertices is \(N\);
  • \(0\) is not written side by side,

so hereinafter we will count the number of binary trees that satisfies the conditions above.

First let us consider DP (Dynamic Programming). For \(i \gt 0\), let

  • \(a_i :=\) (the number of beautiful binary trees of degree \(i\));
  • \(b_i :=\) (the number of beautiful binary trees such that the root has \(0\) written on it and the sum of values of vertices is \(i\)).

By dividing into cases depending on the sum of values on vertices in the subtree under a children, let us find the relationships between \(a\) and \(b\). First, \(b_i\) can be represented with \(a\) as

\[ b_i = 2 a_i + \sum_{0 \lt j \lt i} a_j a_{i-j}\]

(where the former term correspond to the case where it has one children, and the latter to the case where it has two.)

Similarly, \(a_i\) can be expressed with \(a,b\) as

\[ \begin{aligned} a_1 &= 1 \\ a_i &= 2 (a_{i-1} + b_{i-1}) + \sum_{0 \lt j \lt i-1} (a_j + b_j) (a_{i-1-j} + b_{i-1-j}) & (i \gt 1). \end{aligned} \]

If we compute \(a_i\) and \(b_i\) with these expressions in the increasing order of \(i\), the problem can be solved in a total of \(\mathrm{O}(N^2)\) time. (With a combination with divide-and-conquer FFT (Fast Fourier Transform), it can be sped up to \(\mathrm{O}(N \log^2 N)\) time.)

Generating Function

For even better optimization, we introduce generating function. A generating function refers to a formal power series that stores information about a sequence.

  • A formal power series is a generalization of polynomials whose number of terms is not necessarily finite.

For example, for a sequence \(a_1,a_2,\dots\), the ordinary generating function \(A(x)\) of \(a_n\) is defined as

\[ A(x) = a_0 + a_1 x +a_2 x^2 + \dots = \sum_{i=0}^{\infty} a_i x^i. \]

Also, we define a notation \(\lbrack x^n \rbrack\) of extracting a coefficient of a formal power series as

\[\lbrack x^n \rbrack A(x) := a_n.\]

Let \(A(x)\) and \(B(x)\) be the generating functions of \(a_i\) and \(b_i\), respectively. With an aid of generating functions, the aforementioned recurrence relations can be concisely expressed as a convolution. First, \(B(x)\) is expressed with \(A(x)\) as

\[ \begin{aligned} b_i &= 2 a_i + \sum_{0 \lt j \lt i} a_j a_{i-j} \\ &\iff B = 2 A + A^2. \end{aligned} \]

(For simplicity, we adopt a shorthand like \(B(x) \to B\).) Similarly, \(A(x)\) can be written as

\[ \begin{aligned} a_1 &= 1 \\ a_i &= 2 (a_{i-1} + b_{i-1}) + \sum_{0 \lt j \lt i-1} (a_j + b_j) (a_{i-1-j} + b_{i-1-j}) & (i \gt 1) \\ &\iff A = x + 2 x A + 2 x B + x(A + B)^2 \\ &\iff A = x(1 + A + B)^2 = x(1 + 3A + A^2)^2 \end{aligned} \]

  • In the explanation above, we found \(A(x)\) through the recurrence relations, but more simply, it can be found as “the child’s generating function is \((1 + A + B)\), so \(A = x (1 + A + B)^2\).”

Therefore, the desired answer can be obtained as the \(N\)-th coefficient of \(A(x)\) satisfying

\[ A(x) = x(1 + 3A(x) + A(x)^2)^2. \]

With this relations, we can obtain the answer in an \(\mathrm{O}(N \log N)\) time (with Newton’s algorithm; search it if you are interested in.), but this is still not enough for the TL (Time Limit).

Lagrange inversion theorem

Lagrange inversion theorem is a strong weapon for tree-counting problems using generating functions.

Lagrange inversion theorem is a formula of, given a function \(F(x)\), finding the coefficients of the inverse function \(G(x)\) satisfying \(G(F(x))=x\). Among several expression of the theorem, we will show the one that is often used for combinatorics.

Assume that two formal power series \(F(x)\) and \(G(x)\) satisfies \(F(G(x)) = x\). If \(\lbrack x^0 \rbrack F(x)=\lbrack x^0 \rbrack G(x)=0\) and \(\lbrack x^1 \rbrack F(x) \neq 0, \lbrack x^1 \rbrack G(x) \neq 0\), then

\[\lbrack x^n \rbrack G(x) = \frac{1}{n} \lbrack x^{n-1} \rbrack \left(\frac{x}{F(x)}\right)^n.\]

Proof Hereinafter, we use formal Laurent series (a formal power series with finite number of negative-power terms). First we prove the following Lemma.

Lemma (Formal residue)

For any formal power series $F(x)$ such that $F(0)=0, F'(0) \neq 0$, and any integer $k \in \Z$, the following equation holds. $$\lbrack x^{-1} \rbrack F'(x) F^k (x) = \lbrack k = -1\rbrack$$ (Proof of the lemma)

If $k \neq -1$, then it is obvious because $F'(x) F^k (x) = \frac{1}{k+1}\left(F(x)^{k+1}\right)'$.
If $k = -1$, let $F(x) = \sum_{i>0} a_i x^i$, then $$\frac{F'(x)}{F(x)} = x^{-1} \frac{1 + 2\frac{a_2}{a_1}x + \cdots}{1 + \frac{a_2}{a_1}x + \cdots}$$, so the coefficient of $(-1)$-th degree for $k = -1$ is $1$.$\Box$

By the lemma, for $F(x)$ and $G(x)$ satisfying $F(G(x)) = x, F(0)=0, F'(0) \neq 0$, $$F(G(x)) = x.$$ By differentiating both hand sides, $$F'(G) \cdot G' = 1.$$ By decomposing the $F'(G)$ part by coefficients, $$\sum_{i} i (\lbrack x^i \rbrack F (x)) G^{i-1} G' = 1.$$ By multiplying $G^{-n}$ to both hand sides, $$\sum_{i} i (\lbrack x^i \rbrack F (x)) G^{i-1-n} G' = G^{-n}.$$ By extracting $\lbrack x^{-1} \rbrack$ from both hand sides, $$\lbrack x^{-1} \rbrack \sum_{i} i (\lbrack x^i \rbrack F (x)) G^{i-1-n} G' = \lbrack x^{-1} \rbrack G^{-n}$$ By the lemma, $$n \lbrack x^n \rbrack F = \lbrack x^{-1} \rbrack G^{-n}$$ The desired equation can be transformed from this equation.

Example: Catalan number

We will first find the general term of Catalan numbers, \(1,1,2,5,14,\dots\), with the inversion theorem.

Let \(C(x)\) be the generating function of Catalan numbers \(c_i\). Let \(F(x)=C(x)-1\), then the relations

\[F(x) = x(F(x) + 1)^2\]

holds (which derives from the recurrence relations), so for

\[G(x) = \frac{x}{(x+1)^2},\]

\(G(F(x)) = x\) holds. Therefore

\[ \begin{aligned} [x^n]F(x) &= \frac{1}{n} \lbrack x^{n-1} \rbrack \left(\frac{x}{G(x)}\right)^n \\ &= \frac{1}{n} \lbrack x^{n-1} \rbrack (x + 1)^{2n} \\ &= \frac{1}{n} \binom{2n}{n-1} = \frac{1}{n+1} \binom{2n}{n}, \end{aligned} \]

thus, with \(c_0 = 1\), we could obtain

\[c_n = \frac{1}{n+1} \binom{2n}{n}.\]

Now let us adopt the inversion formula to the original problem. Again, what we want is the \(N\)-th coefficient of \(F(x)\) that satisfies

\[F(x) = x(1 + 3F(x) + F(x)^2)^2.\]


\[G(x) := \frac{x}{(1+3x+x^2)^2},\]

then \(G(x)\) is the inverse function of \(F(x)\), so by assigning it in the Lagrange inversion theorem

\[ \lbrack x^n \rbrack F(x) = \frac{1}{n} \lbrack x^{n-1} \rbrack \left( \frac{x}{G(x)} \right)^n, \]

we can obtain

\[ \lbrack x^N \rbrack F(x) = \frac{1}{n} [x^{N-1}] (1+3x+x^2)^{2N}. \]

All that left is to find the \((N-1)\)-th coefficient of \(U(x) := T(x)^m\) for \(T(x) := 1+3x+x^2\) and \(m := 2N\).
At a glance, it seems that computing \(U\) requires \(\mathrm{O}(N \log N)\) time, but with a technique using differentials, we can find the coefficients of \(U\) in an \(\mathrm{O}(N)\) time.

First, by the relations between \(T\) and \(U\), we obtain

\[U' = m T^{m-1} T' \to T U' = m T' U.\]

Let \(u_k := \lbrack x^k \rbrack U(x)\), and by comparing the \((k-1)\)-th coefficient, we obtain

  • left hand side: \(k u_k + 3 (k-1) u_{k-1} + (k-2) u_{k-2}\)
  • right hand side: \(m(3 u_{k-1} + 2 u_{k-2})\)

so the recurrence relations

\[u_k = \frac{3(m + 1 - k)u_{k-1} + (2m + 2 - k)u_{k-2}}{k}\]

hold. Therefore, starting from the initial term \(u_0 = 1\), and using the recurrence relations and the enumeration of inverses, we can compute \(u_{N-1}\) in a linear time. Therefore, the problem has been solved in a total of \(\mathrm{O}(N)\) time. (Since \(u_n\) is P-recursive, high-level algorithm enables us to compute them in an \(\mathrm{O}(\sqrt{N} \log N)\) time, whose detail will be appended in a few hours.)

P.S. I found in a participant’s submission that \(u_k\) can also be computed with a method using \(u_k\).

\[ \begin{aligned} u_k &= \lbrack x^k \rbrack (1+3x+x^2)^{2N} \\ &= \lbrack x^k \rbrack ( (1+3x)+x^2)^{2N} \\ &=\lbrack x^k \rbrack \sum_{0 \leq j \leq 2N} \binom{2N}{j}x^{2j} (1+3x)^{2N-j} \\ &=\sum_{0 \leq j \leq 2N} \binom{2N}{j}\lbrack x^{k-2j} \rbrack (1+3x)^{2N-j} \\ &=\sum_{0 \leq j \leq 2N} \binom{2N}{j} \binom{2N-j}{k-2j} 3^{k-2j}, \end{aligned} \]

so the \(k\)-th term of \((u_n)\) has been computed in an \(\mathrm{O}(N)\) time.

Another solution: P-recursive

There is an alternative way of using a property of the sequence called P-recursive. A sequence \(a_n\) is said to be P-recursive if there exists polynomials on \(n\), \(p_0 (n), p_1(n), \dots\) and \(p_r(n)\), such that for all \(n \geq 0\),

\[ p_0(n) a_n + p_1(n) a_{n+1} + \dots + p_r(n) a_{n+r} = 0.\]

In terms of generating function, let \(A(x)\) be the ordinary generative function of \(a_n\), then P-recursive is defined by

\[ Q_0(x) A(x) + Q_1(x) A'(x) + \dots + Q_r(x) A^{(r)}(x) = 0\]

Examples of P-recursive sequence are factorials, Catalan numbers, and Montmort Numbers.

The generating function for this problem was \(F(x)\) such that

\[F = x(1 + 3F + F^2)^2.\]

By transforming the expression above, we can see that there exists \(Q_0(x),Q_1(x),Q_2(x)\), and \(Q_3(x)\) such that

\[ Q_0(x)F + Q_1(x) F' + Q_2(x) F'' + Q_3(x) F''' = 0,\]

so we can see that \(F(x)\) is a generative function of a P-recursive sequence.

We will explain how to find the \(n\)-th term of coefficients of P-recursive sequence. First, let \(\displaystyle d:= \max_{0 \leq i \leq r}\deg (p_i(n))\).

First of all, we will compute \(p_0(n),\dots,p_r(n)\). To find them, we first naively compute the first \(((r + 1)(d + 2) - 2)\) terms of \(a_n\), and derive a simultaneous linear equations in which the coefficients sequence is regarded as a variable. Then the simultaneous equations can be solved in a total of \(\mathrm{O}((rd)^3)\) to reconstruct them.
After reconstruction, all that left is to compute \(a_n\) in an \(\mathrm{O}(ndr)\) time with the recurrence relations, or in an \(\mathrm{O}(\sqrt{nd}(r^3 + r^2 \log(nd)))\) time using an advanced algorithm consisting of doubling, Lagrange interpolation, and convolutions.

In this problem, \(n = N, d = r = 3 = \mathrm{const.}\), so the problem has been solved in a total of \(\mathrm{O}(N)\) or \(\mathrm{O}(\sqrt{N} \log N)\) time.

The following is a sample implementation in PyPy of the solution using the inversion theorem.

N = int(input())
mod, m, u, inv = 998244353, N * 2, [1] + [0] * (N + 10), [1] * (N + 10)
for k in range(1, N):
  u[k] = (u[k - 1] * 3 * (m + 1 - k) + u[k - 2] * (2 * m + 2 - k)) % mod * inv[k] % mod
  inv[k + 1] = mod - inv[mod % (k + 1)] * (mod // (k + 1)) % mod
print(u[N - 1] * inv[N] % mod)

Similar problems

Several related problems have appeared in contests, mainly in CF (CodeForces) Div.1. If you are interested, we recommend you to try solving them.

  • CF 438E
    • When it comes to counting binary trees using generating functions in programming contests, this is one of the most famous ones. The difficulty is suitable also for practicing Newton’s algorithm and verifications of library.
  • CF 1349F2
    • This is a problem written by Elegia, who invented many techniques using generating functions. Both the observation part and computation part is extremely difficult. The latter half of the observation uses the technique of using Lagrange inversion theorem.
  • TCO19 *** (avoiding spoiler)
    • The writer’s solution is an orthodox combinatorics, but some contestants is said to have solved it with P-recursive. I have an impression that quite a few problems can be solved with P-recursive, although unintended.
  • CF 1479E
    • The writer’s solution requires the algorithm of computing P-recursive sequence in an \(\mathrm{O}(\sqrt{N} \log N)\) time. The problem reflects nowadays’ trend of combinatorics, including the former half observation of using martingale.

Related articles

We will introduce some recommended texts and articles related to this problem.

  • maspy, 形式的べき級数解説 (in Japanese)
    • If you are a Japanese who want to learn formal power series, this is by far the best article to dive into. Explained in plain words, it is very easy to understand, and moreover there are many exercises, so beginners will benefit a lot in studying.
  • 37zigen, 指数型母関数入門 (in Japanese)
    • Although not mentioned in this editorial, other than ordinary generating function, there is another important concept called exponential generating function. This article explains exponential generating function, with many problems in contests, so this is very informative.
  • zscoder, Generating Functions in Competitive Programming (Part 2)
    • This explains competitive programming problems using generating functions thoroughly. While Part 1 is enlightening too, Part 2 has many editorials of problems using Lagrange inversion theorem, which are more related to this problem.
  • Retired_MiFaFaOvO, A problem collection of ODE and differential technique
    • This is an article written by apiad, who is a golden-crown ranker of AtCoder. It explains the technique of using differentials and P-recursive, which appear in this problem.
  • Elegia, Athekatelan, 信息学竞赛中的生成函数计算理论框架 (in Chinese)
    • The title is translated like “The Structure of Computation Theory of Generating Functions in Programming Contest”. It summarizes state-of-the-art techniques released from China in the last couple of years, including techniques using Lagrange inversion theorem and differntials.
  • noshi91, polynomial_matrix_prod.rs
    • This is a sample implementation of algorithms required to find the \(N\)-th term of a P-recursive sequence in a total of \(\mathrm{O}(\sqrt{N} \log N)\). For a matrix with polynomial coefficients \(M(n)\), if the expressions like \( M(N) \times M(N-1) \times \dots \times M(2) \times M(1)\) can be computed fast, then so can P-recursive. noshi’s library is an implementation of this.

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