查看“︁赋权二叉树”︁的源代码

赋权二叉树（Weighted Binary Tree）是FataliS1024提出的大数函数。

=== 定义 ===
对于有根二叉树，令其每条边都有一个正整数权值，即得到赋权二叉树，记作'''wb'''

对于两个'''wb''' A和B，如果A能通过以下操作得到B，就称B嵌入A，A容纳B，A大于B，B小于A：

# 删掉一个度为1的顶点和它连接的边
# 删掉一个度为2的非根顶点和它连接的两条边，并将它原本连接的两个顶点连起来，权值等于min(原来两条边的权值)
# 将任意一个大于1的权值-1

符合以下条件的最长的有序wb列的长度记作wbtree(n)：

# 第k个wb最多有k+1个顶点
# 所有wb的边权值不超过n
# 前面的wb不小于后面的wb

=== 分析 ===
用()表示权值1的边与它的子节点（远离根的一端）。用[]表示权值2。{}表示权值3.根节点不写

以下提供了wbtree中的一个序型分析

* 单根=0
* <math>() = 1</math>
* <math>(()) = 2</math>
* <math>()() = \omega</math>

上述这些都跟ε(0)以下的tree相同

<math>\begin{align}.\\&[] = \varepsilon_0\\&([]) = \varepsilon_0+1\\&(([])) = \varepsilon_0+2\\&([])() = \varepsilon_0+\omega\\&([])([]) = \varepsilon_0\times2\\&(([]))([]) = \varepsilon_0\times3\\&(([])())([]) = \varepsilon_0\times\omega\\&(([])([]))([]) = \varepsilon_0^2\\&(([]))(([])) = \varepsilon_0^\omega\\&(([])([]))(([])([])) = \varepsilon_0^\varepsilon_0\\&[]() = \varepsilon_1\\&[](()) =  \varepsilon_2 \\&[](()()) =  \varepsilon_\omega\\&[]([]) =  \varepsilon_{\varepsilon_0}\\&[]([]([])) =  \varepsilon_{\varepsilon_{\varepsilon_0}}\\&[()] = \zeta_0\\&([()]) = \zeta_0+1\\&([()])([()]) = \zeta_0\times2\\&[]([()]) = \varepsilon_{\zeta_0+1}\\&[](([()])) = \varepsilon_{\zeta_0+2}\\&[]([]([()])) = \varepsilon _{\varepsilon_{\zeta_0+1}}\\&[()]() = \zeta_1\\&[()]([()]) = \zeta_{\zeta_0}\\&[(())] = \vartheta(\Omega\times3) = \varphi(3,0)\\&[]([(())]) = \vartheta(\Omega+\vartheta(\Omega\times3))\\&[()]([(())]) = \vartheta(\Omega\times2+\vartheta(\Omega\times3))\\&[(())]() = \vartheta(\Omega\times3+1)\\&[((()))] = \vartheta(\Omega\times4)\\&[()()] = \vartheta(\Omega\times\omega)\\&[(()())] = \vartheta(\Omega\times(\omega+1))\\&[(()())(()())] = \vartheta(\Omega\times\omega^{\omega^{\omega}})\\&[([])] = \vartheta(\Omega\times\vartheta(\Omega))\\&[([])]() = \vartheta(\Omega\times\vartheta(\Omega)+1)\\&[(([]))] = \vartheta(\Omega\times(\vartheta(\Omega)+1))\\&[([])([])] = \vartheta(\Omega\times\vartheta(\Omega)\times2)\\&[([]())] = \vartheta(\Omega\times\vartheta(\Omega+1))\\&[([()])] = \vartheta(\Omega\times\vartheta(\Omega\times2))\\&[([([])])] = \vartheta(\Omega\times\vartheta(\Omega\times\vartheta(\Omega)))\\&[[]] = \vartheta(\Omega^2) =\Gamma_0\\&[]([[]]) = \vartheta(\Omega+\vartheta(\Omega^2))\\&[()]([[]]) = \vartheta(\Omega\times2+\vartheta(\Omega^2))\\&[([[]])] = \vartheta(\Omega\times\vartheta(\Omega^2))\\&[([([[]])])] = \vartheta(\Omega\times\vartheta(\Omega\times\vartheta(\Omega^2)))\\&[[]]() = \vartheta(\Omega^2+1)\\&[[]]([[]]) = \vartheta(\Omega^2+\vartheta(\Omega^2))\\&[[]()] = \vartheta(\Omega^2+\Omega)\\&[[]([[]])] = \vartheta(\Omega^2+\Omega\times\vartheta(\Omega^2))\\&[[()]] = \vartheta(\Omega^2\times2)\\&[[([[]])]] = \vartheta(\Omega^2\times\vartheta(\Omega^2))\\&[[[]]] = \vartheta(\Omega^3)\\&[[[]]]() = \vartheta(\Omega^3+1)\\&[[[]]()] = \vartheta(\Omega^3+\Omega)\\&[[[]()]] = \vartheta(\Omega^3+\Omega^2)\\&[[[()]]] = \vartheta(\Omega^3\times2)\\&[[[[]]]] = \vartheta(\Omega^4)\\&[][] = \vartheta(\Omega^\omega)\\&([][]) = \vartheta(\Omega^\omega)+1\\&[]([][]) = \vartheta(\Omega+\vartheta(\Omega^\omega))\\&[([][])] = \vartheta(\Omega\times\vartheta(\Omega^\omega))\\&[[([][])]] = \vartheta(\Omega^2\times\vartheta(\Omega^\omega))\\&[[][]] = \vartheta(\Omega^\omega+1)\\&[[][]]() = \vartheta(\Omega^\omega+2)\\&[[][]]([][]) = \vartheta(\Omega^\omega+\vartheta(\Omega^\omega))\\&[[[][]]] = \vartheta(\Omega^\omega+\Omega)\\&[[[][]]]() = \vartheta(\Omega^\omega+\Omega+1)\\&[[[][]]()] = \vartheta(\Omega^\omega+\Omega\times2)\\&[[[[][]]]] = \vartheta(\Omega^\omega+\Omega^2)\\&[[[[[][]]]]] = \vartheta(\Omega^\omega+\Omega^3)\\&[()][] = \vartheta(\Omega^\omega\times2)\\&[[()][]] = \vartheta(\Omega^\omega\times2+1)\\&[[[()][]]] = \vartheta(\Omega^\omega\times2+\Omega)\\&[[[[()][]]]] = \vartheta(\Omega^\omega\times2+\Omega^2)\\&[(())][] = \vartheta(\Omega^\omega\times3)\\&[([][])][] = \vartheta(\Omega^\omega\times\vartheta(\Omega^\omega))\\&[[]][] = \vartheta(\Omega^{\omega+1})\\&[[[]]][] = \vartheta(\Omega^{\omega+2})\\&[[][]][] = \vartheta(\Omega^{\omega\times2})\\&[[[][]][]] = \vartheta(\Omega^{\omega\times2}+1)\\&[[[[][]][]]] = \vartheta(\Omega^{\omega\times2}+\Omega)\\&[[[][]]][] = \vartheta(\Omega^{\omega\times2}+\Omega^\omega)\\&[[[][]]()][] = \vartheta(\Omega^{\omega\times2}+\Omega^\omega\times2)\\&[[[[][]]]][] = \vartheta(\Omega^{\omega\times2}+\Omega^{\omega+1})\\&[[()][]][] = \vartheta(\Omega^{\omega\times2}\times2)\\&[[[]][]][] = \vartheta(\Omega^{\omega\times2+1})\\&[[[][]][]][] = \vartheta(\Omega^{\omega\times3})\\&[[[[][]][]][]][] = \vartheta(\Omega^{\omega\times4})\\&[()][()] = \vartheta(\Omega^{\omega^2})\\&[[()][()]] = \vartheta(\Omega^{\omega^2}+1)\\&[[[()][()]][]] = \vartheta(\Omega^{\omega^2}+\Omega^\omega)\\&[(())][()] = \vartheta(\Omega^{\omega^2}\times2)\\&[[]][()] = \vartheta(\Omega^{\omega^2+1})\\&[[()][()]][()] = \vartheta(\Omega^{\omega^2\times2})\\&[(())][(())] = \vartheta(\Omega^{\omega^3})\\&[([])][([])] = \vartheta(\Omega^{\vartheta(\Omega)})\\&[[]][[]] = \vartheta(\Omega^\Omega) = LVO\end{align}</math>

看上去[]可以作为Ω的角色了，这样只使用权重1~2就能达到至少BHO的序型

<math>\begin{align}.\\&\psi(\Omega_2)=\{\}\\&\psi(\Omega_2+\Omega)=[\{\}]\\&\psi(\Omega_2+\Omega^\Omega^\omega)=[\{\}][]\\&\psi(\Omega_2+\psi_1(\Omega_2))=[\{\}][\{\}]\\&\psi(\Omega_2+\psi_1(\Omega_2)+\Omega)=[\{\}][[\{\}]]\\&\psi(\Omega_2+\psi_1(\Omega_2)\times2)=[\{\}][[\{\}][\{\}]]\\&\psi(\Omega_2+\psi_1(\Omega_2)\times3)=[\{\}][[\{\}][[\{\}][\{\}]]]\\&\psi(\Omega_2+\psi_1(\Omega_2+1))=[[\{\}]][[\{\}]]\\&\psi(\Omega_2+\psi_1(\Omega_2+2))=[[[\{\}]]][[[\{\}]]]\\&\psi(\Omega_2+\psi_1(\Omega_2+\omega))=[[\{\}]()][[\{\}]()]\\&\psi(\Omega_2+\psi_1(\Omega_2+\Omega))=[[\{\}][]][[\{\}][]]\\&\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))=[[\{\}][\{\}]][[\{\}][\{\}]]\\&\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2))))=[[[\{\}][\{\}]][[\{\}][\{\}]]][[[\{\}][\{\}]][[\{\}][\{\}]]]\\&\psi(\Omega_2\times2)=\{\}()\\&\psi(\Omega_2\times\omega)=\{\}(()())\\&\psi(\Omega_2\times\Omega)=\{\}[]\\&\psi(\Omega_2\times\psi_1(\Omega_2))=\{\}[\{\}]\\&\psi(\Omega_2\times\psi_1(\Omega_2\times\psi_1(\Omega_2)))=\{\}[\{\}[\{\}]]\\&\psi(\Omega_2^2=\{()\}\\&\psi(\Omega_2^\omega)=\{()()\}\\&\psi(\Omega_2^\Omega)=\{[]\}\\&\psi(\Omega_2^{\Omega_2})=\{\{\}\}\\&\psi(\Omega_2^{\Omega_2^2})=\{\{\{\}\}\}\\&\psi(\Omega_2^{\Omega_2^3})=\{\{\{\{\}\}\}\}\\&\psi(\Omega_2^{\Omega_2^\omega})=\{\}\{\}\\&\psi(\Omega_2^{\Omega_2^\Omega})=\{[]\}\{[]\}\\&\psi(\Omega_2^{\Omega_2^{\Omega_2}})=\{\{\}\}\{\{\}\}\\&\psi(\Omega_2^{\Omega_2^{\Omega_2^\omega}})=\{\{\}\{\}\}\{\{\}\{\}\}\\&\psi(\Omega₃)=\cdots\end{align}</math>

<nowiki>\begin{align}.\\&\psi(\Omega_2)=\{\}\\&\psi(\Omega_2+\Omega)=[\{\}]\\&\psi(\Omega_2+\Omega^\Omega^\omega)=[\{\}][]\\&\psi(\Omega_2+\psi_1(\Omega_2))=[\{\}][\{\}]\\&\psi(\Omega_2+\psi_1(\Omega_2)+\Omega)=[\{\}][[\{\}]]\\&\psi(\Omega_2+\psi_1(\Omega_2)\times2)=[\{\}][[\{\}][\{\}]]\\&\psi(\Omega_2+\psi_1(\Omega_2)\times3)=[\{\}][[\{\}][[\{\}][\{\}]]]\\&\psi(\Omega_2+\psi_1(\Omega_2+1))=[[\{\}]][[\{\}]]\\&\psi(\Omega_2+\psi_1(\Omega_2+2))=[[[\{\}]]][[[\{\}]]]\\&\psi(\Omega_2+\psi_1(\Omega_2+\omega))=[[\{\}]()][[\{\}]()]\\&\psi(\Omega_2+\psi_1(\Omega_2+\Omega))=[[\{\}][]][[\{\}][]]\\&\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))=[[\{\}][\{\}]][[\{\}][\{\}]]\\&\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2))))=[[[\{\}][\{\}]][[\{\}][\{\}]]][[[\{\}][\{\}]][[\{\}][\{\}]]]\\&\psi(\Omega_2\times2)=\{\}()\\&\psi(\Omega_2\times\omega)=\{\}(()())\\&\psi(\Omega_2\times\Omega)=\{\}[]\\&\psi(\Omega_2\times\psi_1(\Omega_2))=\{\}[\{\}]\\&\psi(\Omega_2\times\psi_1(\Omega_2\times\psi_1(\Omega_2)))=\{\}[\{\}[\{\}]]\\&\psi(\Omega_2^2=\{()\}\\&\psi(\Omega_2^\omega)=\{()()\}\\&\psi(\Omega_2^\Omega)=\{[]\}\\&\psi(\Omega_2^{\Omega_2})=\{\{\}\}\\&\psi(\Omega_2^{\Omega_2^2})=\{\{\{\}\}\}\\&\psi(\Omega_2^{\Omega_2^3})=\{\{\{\{\}\}\}\}\\&\psi(\Omega_2^{\Omega_2^\omega})=\{\}\{\}\\&\psi(\Omega_2^{\Omega_2^\Omega})=\{[]\}\{[]\}\\&\psi(\Omega_2^{\Omega_2^{\Omega_2}})=\{\{\}\}\{\{\}\}\\&\psi(\Omega_2^{\Omega_2^{\Omega_2^\omega}})=\{\{\}\{\}\}\{\{\}\{\}\}\\&\psi(\Omega₃)=\cdots\end{align}</nowiki>

于是得到ψ(Ω_ω)=极限
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