F - Intervals on Tree

Score : $600$ points

Problem Statement

We have a tree with $N$ vertices and $N-1$ edges, respectively numbered $1, 2,\cdots, N$ and $1, 2, \cdots, N-1$. Edge $i$ connects Vertex $u_i$ and $v_i$.

For integers $L, R$ ($1 \leq L \leq R \leq N$), let us define a function $f(L, R)$ as follows:

• Let $S$ be the set of the vertices numbered $L$ through $R$. $f(L, R)$ represents the number of connected components in the subgraph formed only from the vertex set $S$ and the edges whose endpoints both belong to $S$.

Compute $\sum_{L=1}^{N} \sum_{R=L}^{N} f(L, R)$.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq u_i, v_i \leq N$
• The given graph is a tree.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$

$u_2$ $v_2$

$:$

$u_{N-1}$ $v_{N-1}$

Output

Print $\sum_{L=1}^{N} \sum_{R=L}^{N} f(L, R)$.

3
1 3
2 3

7

We have six possible pairs $(L, R)$ as follows:

• For $L = 1, R = 1$, $S = \{1\}$ and we have $1$ connected component.
• For $L = 1, R = 2$, $S = \{1, 2\}$ and we have $2$ connected components.
• For $L = 1, R = 3$, $S = \{1, 2, 3\}$ and we have $1$ connected component, since $S$ contains both endpoints of each of the edges $1, 2$.
• For $L = 2, R = 2$, $S = \{2\}$ and we have $1$ connected component.
• For $L = 2, R = 3$, $S = \{2, 3\}$ and we have $1$ connected component, since $S$ contains both endpoints of Edge $2$.
• For $L = 3, R = 3$, $S = \{3\}$ and we have $1$ connected component.

The sum of these is $7$.

2
1 2

3

10
5 3
5 7
8 9
1 9
9 10
8 4
7 4
6 10
7 2

113