# 数学代写|组合学代写Combinatorics代考|Unlabeled Full Binary RP-Trees

#### Doug I. Jones

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## 数学代写|组合学代写Combinatorics代考|Unlabeled Full Binary RP-Trees

We’ll begin with a review of material discussed in Examples 7.9 (p. 206) and 7.10. Roughly speaking, an unlabeled RP-tree is an RP-tree with the vertex labels erased. Thus, the order of the sons of a vertex is still important. A tree is “binary” (resp. “full binary”) if each nonleaf has at most (resp. exactly) two sons. Figure 9.5 shows some unlabeled full binary RP-trees. Here is a more precise pictorial definition. Compare it to Definition 9.1 for (labeled) RP-trees.

Definition 9.4 Unlabeled binary rooted plane trees The following are unlabeled binary RP-trees. Roots are indicated by $\bullet$ and other vertices by $\circ$.
(i) The single vertex $\bullet$ is such a tree.
(ii) If $T_1$ is one such tree, so is the tree formed by (a) drawing $T_1$ root upward, (b) adding a above $T_1$ and connecting $\bullet$ to the root of $T_1$, and (c) changing the root of $T_1$ to $\circ$.
(iii) If $T_1$ and $T_2$ are two such trees, so is the tree formed by (a) drawing $T_1$ to the left of $T_2$, both root upward, (b) adding a $\bullet$ above them and connecting it to their roots, and (c) changing the roots of $T_1$ and $T_2$ to o’s.
If we omit (ii), the result is unlabeled full binary RP-trees.
These trees are often referred to as “unlabeled ordered (full) binary trees.” Why? To define a binary tree, one needs to have a root. Drawing a tree in the plane is equivalent to ordering the children of each vertex. Sometimes the adjective “full” is omitted. In this section, we’ll study unlabeled ordered full binary trees.

We can build all unlabeled full binary RP-trees recursively by applying the definition over and over. To begin with there are no trees, so all we get is a single vertex by (1) of the definition. This tree can then be used to build the 3 vertex full binary RP-tree shown in the next step of Figure 9.5 . Using the two trees now available, we can build the three new trees shown in the right hand step of Figure 9.5. In general, if we have a total of $t_n$ trees at step $n$, then $t_1=1$ (the single vertex) and $t_{n+1}=1+\left(t_n\right)^2+1$ (either use the single vertex tree or join two trees $T_1$ and $T_2$ to a new root).

## 数学代写|组合学代写Combinatorics代考|What are Generating Functions?

In this section, we introduce the idea of ordinary generating functions and look at some ways to manipulate them. This material is essential for understanding later material on generating functions. Be sure to work the exercises in this section before reading later sections!

Definition 10.1 Ordinary generating function (OGF) Suppose we are given a sequence $a_0, a_1, \ldots$. The ordinary generating function (also called OGF) associated with this sequence is the function whose value at $x$ is $\sum_{i=0}^{\infty} a_i x^i$. The sequence $a_0, a_1, \ldots$ is called the coefficients of the generating function.

People often drop “ordinary” and call this the generating function for the sequence. This is also called a “power series” because it is the sum of a series whose terms involve powers of $x$. The summation is often written $\sum_{i \geq 0} a_i x^i$ or $\sum a_i x^i$.

If your sequence is finite, you can still construct a generating function by taking all the terms after the last to be zero. If you have a sequence that starts at $a_k$ with $k>0$, you can define $a_0, \ldots, a_{k-1}$ to be any convenient values. “Convenient values” are ones that make equations nicer in some sense. For example, if $H_{n+1}=2 H_n+1$ for $n>0$ and $H_1=1$. It is convenient to let $H_0=0$ so that the recursion is valid for $n \geq 0$. ( $H_n$ is the number of moves required for the Tower of Hanoi puzzle. See Exercise 7.3.9 (p. 218).) On the other hand, if $b_1=1$ and $b_n=\sum_{k=1}^{n-1} b_k b_{n-k}$ for $n>1$, it’s convenient to define $b_0=0$ so that we have $b_n=\sum_{k=0}^n b_k b_{n-k}$ for $k \neq 1$. (The latter sum is a “convolution”, which we will define in a little while.)

To help us keep track of which generating function is associated with which sequence, we try to use lower case letters for sequences and the corresponding upper case letters for the generating functions. Thus we use the function $A$ as generating function for a sequence of $a_n$ ‘s and $B$ as the generating function for $b_n$ ‘s. Sometimes conventional notation for certain sequences make this upper and lower case pairing impossible. In those cases, we improvise.

You may have noticed that our definition is incomplete because we spoke of a function but did not specify its domain or range. The domain will depend on where the power series converges; however, for combinatorial applications, there is usually no need to be concerned with the convergence of the power series. As a result of this, we will often ignore the issue of convergence. In fact, we can treat the power series like a polynomial with an infinite number of terms. The domain in which the power series converges does matter when we study asymptotics, but that is still several sections in the future.

If we have a doubly indexed sequence $b_{i, j}$, we can extend the definition of a generating function:
$$B(x, y)=\sum_{j \geq 0} \sum_{i \geq 0} b_{i, j} x^i y^j=\sum_{i, j=0}^{\infty} b_{i, j} x^i y^j .$$
Clearly, we can extend this idea to any number of indices-we’re not limited to just one or two.

# 组合学代考

## 数学代写|组合学代写Combinatorics代考|Unlabeled Full Binary RP-Trees

9.4未标记二叉有根平面树以下为未标记二叉rp树。根由$\bullet$表示，其他顶点由$\circ$表示。
(i)单顶点$\bullet$就是这样一棵树。
(ii)如果$T_1$是这样一棵树，那么(a)将$T_1$根向上绘制，(b)在$T_1$上面添加a并将$\bullet$连接到$T_1$的根，以及(c)将$T_1$的根更改为$\circ$所形成的树也是如此。
(iii)如果$T_1$和$T_2$是两棵这样的树，那么(a)将$T_1$画在$T_2$的左边，两个根都向上，(b)在它们上面加上一个$\bullet$并将其连接到它们的根，以及(c)将$T_1$和$T_2$的根改为o的树也是如此。

## 数学代写|组合学代写Combinatorics代考|What are Generating Functions?

$$B(x, y)=\sum_{j \geq 0} \sum_{i \geq 0} b_{i, j} x^i y^j=\sum_{i, j=0}^{\infty} b_{i, j} x^i y^j .$$

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