期望的线性性质：\(\displaystyle\sum E_{x->y}=E_{x->x+1}+E_{x+1->x+2}+...+E_{y-1->y}=\sum_{i=x}^{y-1}E_{i->i+1}\)
设 \(d_i=i\) 的返祖边条数。\(E_i\) 为 \(i\) 的返祖边集。
\(E_{x->x+1}=\dfrac{1}{d_x+1}+\dfrac{1}{d_x+1}\displaystyle\sum_{(x,y)\in E_x}E_{y->x+1}=\dfrac{1}{d_x+1}+\dfrac{1}{d_x+1}\displaystyle\sum_{(x,y)\in E_x}\sum_{i=x}^{y}E_{i->i+1}\)
令 \(f_x=E_{x->x+1},s_x\) 为 \(f\) 的前缀和。
\(f_x=\dfrac{1}{d_x+1}\displaystyle+\dfrac{1}{d_x+1}\sum_{(x,y)\in E_x}(s_x-s_{y-1})=\dfrac{1}{d_x+1}\displaystyle+\dfrac{1}{d_x+1}\sum_{(x,y)\in E_x}(s_{x-1}-s_{y-1})+\dfrac{d_x}{d_x+1}f_x\)
为了使 \(f_x\) 只出现在一边，移项 \(\dfrac{1}{d_x+1}f_x=\dfrac{1}{d_x+1}\displaystyle+\dfrac{1}{d_x+1}\sum_{(x,y)\in E_x}(s_{x-1}-s_{y-1})\)
\(f_x=1+\sum_{(x,y)\in E_x}(s_{x-1}-s_{y-1})\)