什么是树上dp？字面意思，在树上做dp
但是这个描述不准确，更准确应该是树形dp，树上dp可能只是在树上，有一个dp，不一定是树状转移状态的
状态转移的方向：自顶向下/自底向上
自顶向下需要为根节点设计初始状态，自底向上需要为所有叶子节点设计初始状态，这里可能是不同的
一个自顶向下的例子：树上差分/前缀和
自顶向下时一个结点的多个子结点会共用一个dp状态栈，所以需要利用回溯的性质
自底向上时一个结点会从多个子结点来转移状态，所以状态转移方程会更复杂；且子结点树可能是\(O(n)\)的，对于关于子结点数\(k\)时间复杂度\(\Omega(k^{1+\epsilon})\)的转移计算方法，菊花树会爆，此时要考虑在子结点上做一些操作比如排序/在子结点上维护数据结构/启发式合并状态/设计逐结点转移方程/拆点拆边
This article develops the exponential and trigonometric functions from first principles using infinite series, avoiding
reliance on geometric intuition or pre-existing notions of angle. Motivated by foundational concerns raised in early
twentieth-century analysis, the construction is carried out directly on the complex plane. After establishing the
algebraic structure and completeness of
including absolute convergence and a Fubini-type theorem. The exponential function is then defined by its power series,
shown to be well-defined on
monotonicity—are derived. Trigonometric functions are introduced via the complex exponential, leading naturally to
Euler’s formula, addition formulas, and the Pythagorean identity. This approach demonstrates that the classical
properties of exponential and trigonometric functions arise purely from analytic and algebraic considerations, without
geometric assumptions.
\newcommand {\w} {\omega}
\newcommand {\W} {\Omega}
\newcommand {\e} {\varepsilon}
\newcommand {\p} {\psi}
\newcommand {\f} {\varphi}
\newcommand {\z} {\zeta}
\newcommand {\G} {\Gamma}
\newcommand {\a} {\alpha}
Mainly using BOCF with
notation with extended Veblen’s function is also used.
Contents: