赋权二叉树：修订间差异

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2026年2月23日 (一) 19:24的最新版本

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.<>表示权值4.根节点不写。

以下提供了wbtree中的一个序型分析

单根=0
$() = 1$
$(()) = 2$
$() () = ω$

上述这些都跟ε(0)以下的tree相同

$\begin{aligned} . \\ [] = ε_{0} \\ ([]) = ε_{0} + 1 \\ (([])) = ε_{0} + 2 \\ ([]) () = ε_{0} + ω \\ ([]) ([]) = ε_{0} \times 2 \\ (([])) ([]) = ε_{0} \times 3 \\ (([]) ()) ([]) = ε_{0} \times ω \\ (([]) ([])) ([]) = ε_{0}^{2} \\ (([])) (([])) = ε_{0}^{ω} \\ (([]) ([])) (([]) ([])) = ε_{0}^{ε}_{0} \\ [] () = ε_{1} \\ [] (()) = ε_{2} \\ [] (() ()) = ε_{ω} \\ [] ([]) = ε_{ε_{0}} \\ [] ([] ([])) = ε_{ε_{ε_{0}}} \\ [()] = ζ_{0} \\ ([()]) = ζ_{0} + 1 \\ ([()]) ([()]) = ζ_{0} \times 2 \\ [] ([()]) = ε_{ζ_{0} + 1} \\ [] (([()])) = ε_{ζ_{0} + 2} \\ [] ([] ([()])) = ε_{ε_{ζ_{0} + 1}} \\ [()] () = ζ_{1} \\ [()] ([()]) = ζ_{ζ_{0}} \\ [(())] = ϑ (Ω \times 3) = φ (3, 0) \\ [] ([(())]) = ϑ (Ω + ϑ (Ω \times 3)) \\ [()] ([(())]) = ϑ (Ω \times 2 + ϑ (Ω \times 3)) \\ [(())] () = ϑ (Ω \times 3 + 1) \\ [((()))] = ϑ (Ω \times 4) \\ [() ()] = ϑ (Ω \times ω) \\ [(() ())] = ϑ (Ω \times (ω + 1)) \\ [(() ()) (() ())] = ϑ (Ω \times ω^{ω^{ω}}) \\ [([])] = ϑ (Ω \times ϑ (Ω)) \\ [([])] () = ϑ (Ω \times ϑ (Ω) + 1) \\ [(([]))] = ϑ (Ω \times (ϑ (Ω) + 1)) \\ [([]) ([])] = ϑ (Ω \times ϑ (Ω) \times 2) \\ [([] ())] = ϑ (Ω \times ϑ (Ω + 1)) \\ [([()])] = ϑ (Ω \times ϑ (Ω \times 2)) \\ [([([])])] = ϑ (Ω \times ϑ (Ω \times ϑ (Ω))) \\ [[]] = ϑ (Ω^{2}) = Γ_{0} \\ [] ([[]]) = ϑ (Ω + ϑ (Ω^{2})) \\ [()] ([[]]) = ϑ (Ω \times 2 + ϑ (Ω^{2})) \\ [([[]])] = ϑ (Ω \times ϑ (Ω^{2})) \\ [([([[]])])] = ϑ (Ω \times ϑ (Ω \times ϑ (Ω^{2}))) \\ [[]] () = ϑ (Ω^{2} + 1) \\ [[]] ([[]]) = ϑ (Ω^{2} + ϑ (Ω^{2})) \\ [[] ()] = ϑ (Ω^{2} + Ω) \\ [[] ([[]])] = ϑ (Ω^{2} + Ω \times ϑ (Ω^{2})) \\ [[()]] = ϑ (Ω^{2} \times 2) \\ [[([[]])]] = ϑ (Ω^{2} \times ϑ (Ω^{2})) \\ [[[]]] = ϑ (Ω^{3}) \\ [[[]]] () = ϑ (Ω^{3} + 1) \\ [[[]] ()] = ϑ (Ω^{3} + Ω) \\ [[[] ()]] = ϑ (Ω^{3} + Ω^{2}) \\ [[[()]]] = ϑ (Ω^{3} \times 2) \\ [[[[]]]] = ϑ (Ω^{4}) \\ [] [] = ϑ (Ω^{ω}) \\ ([] []) = ϑ (Ω^{ω}) + 1 \\ [] ([] []) = ϑ (Ω + ϑ (Ω^{ω})) \\ [([] [])] = ϑ (Ω \times ϑ (Ω^{ω})) \\ [[([] [])]] = ϑ (Ω^{2} \times ϑ (Ω^{ω})) \\ [[] []] = ϑ (Ω^{ω} + 1) \\ [[] []] () = ϑ (Ω^{ω} + 2) \\ [[] []] ([] []) = ϑ (Ω^{ω} + ϑ (Ω^{ω})) \\ [[[] []]] = ϑ (Ω^{ω} + Ω) \\ [[[] []]] () = ϑ (Ω^{ω} + Ω + 1) \\ [[[] []] ()] = ϑ (Ω^{ω} + Ω \times 2) \\ [[[[] []]]] = ϑ (Ω^{ω} + Ω^{2}) \\ [[[[[] []]]]] = ϑ (Ω^{ω} + Ω^{3}) \\ [()] [] = ϑ (Ω^{ω} \times 2) \\ [[()] []] = ϑ (Ω^{ω} \times 2 + 1) \\ [[[()] []]] = ϑ (Ω^{ω} \times 2 + Ω) \\ [[[[()] []]]] = ϑ (Ω^{ω} \times 2 + Ω^{2}) \\ [(())] [] = ϑ (Ω^{ω} \times 3) \\ [([] [])] [] = ϑ (Ω^{ω} \times ϑ (Ω^{ω})) \\ [[]] [] = ϑ (Ω^{ω + 1}) \\ [[[]]] [] = ϑ (Ω^{ω + 2}) \\ [[] []] [] = ϑ (Ω^{ω \times 2}) \\ [[[] []] []] = ϑ (Ω^{ω \times 2} + 1) \\ [[[[] []] []]] = ϑ (Ω^{ω \times 2} + Ω) \\ [[[] []]] [] = ϑ (Ω^{ω \times 2} + Ω^{ω}) \\ [[[] []] ()] [] = ϑ (Ω^{ω \times 2} + Ω^{ω} \times 2) \\ [[[[] []]]] [] = ϑ (Ω^{ω \times 2} + Ω^{ω + 1}) \\ [[()] []] [] = ϑ (Ω^{ω \times 2} \times 2) \\ [[[]] []] [] = ϑ (Ω^{ω \times 2 + 1}) \\ [[[] []] []] [] = ϑ (Ω^{ω \times 3}) \\ [[[[] []] []] []] [] = ϑ (Ω^{ω \times 4}) \\ [()] [()] = ϑ (Ω^{ω^{2}}) \\ [[()] [()]] = ϑ (Ω^{ω^{2}} + 1) \\ [[[()] [()]] []] = ϑ (Ω^{ω^{2}} + Ω^{ω}) \\ [(())] [()] = ϑ (Ω^{ω^{2}} \times 2) \\ [[]] [()] = ϑ (Ω^{ω^{2} + 1}) \\ [[()] [()]] [()] = ϑ (Ω^{ω^{2} \times 2}) \\ [(())] [(())] = ϑ (Ω^{ω^{3}}) \\ [([])] [([])] = ϑ (Ω^{ϑ (Ω)}) \\ [[]] [[]] = ϑ (Ω^{Ω}) = L V O \end{aligned}$

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

ψ(Ω₂)={}
ψ(Ω₂+Ω)=[{}]
ψ(Ω₂+Ω^Ω^ω)=[{}][]
ψ(Ω₂+ψ₁(Ω₂))=[{}][{}]
ψ(Ω₂+ψ₁(Ω₂)+Ω)=[{}][[{}]]
ψ(Ω₂+ψ₁(Ω₂)×2)=[{}][[{}][{}]]
ψ(Ω₂+ψ₁(Ω₂)×3)=[{}][[{}][[{}][{}]]]
ψ(Ω₂+ψ₁(Ω₂+1))=[[{}]][[{}]]
ψ(Ω₂+ψ₁(Ω₂+2))=[[[{}]]][[[{}]]]
ψ(Ω₂+ψ₁(Ω₂+ω))=[[{}]()][[{}]()]
ψ(Ω₂+ψ₁(Ω₂+Ω))=[[{}][]][[{}][]]
ψ(Ω₂+ψ₁(Ω₂+ψ₁(Ω₂)))=[[{}][{}]][[{}][{}]]
ψ(Ω₂+ψ₁(Ω₂+ψ₁(Ω₂+ψ₁(Ω₂))))=[[[{}][{}]][[{}][{}]]][[[{}][{}]][[{}][{}]]]
ψ(Ω₂×2)={}()
ψ(Ω₂×ω)={}(()())
ψ(Ω₂×Ω)={}[]
ψ(Ω₂×ψ₁(Ω₂))={}[{}]
ψ(Ω₂×ψ₁(Ω₂×ψ₁(Ω₂)))={}[{}[{}]]
ψ(Ω₂²)={()}
ψ(Ω₂^ω)={()()}
ψ(Ω₂^Ω)={[]}
ψ(Ω₂^Ω₂)={{}}
ψ(Ω₂^Ω₂³)={{{{}}}}
ψ(Ω₂^Ω₂^ω)={}{}
ψ(Ω₂^Ω₂^Ω)={[]}{[]}
ψ(Ω₂^Ω₂^Ω₂)={{}}{{}}
ψ(Ω₂^Ω₂^Ω₂^ω)={{}{}}{{}{}}
ψ(Ω₃)=<>
ψ(Ω₄)=…

于是得到ψ(Ω_ω)是wbtree的极限。

@@ 第17行： / 第17行： @@
 === 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
+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.<>表示权值4.根节点不写。
 以下提供了wbtree中的一个序型分析
 * 单根=0
-* () = 1
+* <math>() = 1</math>
-* (()) = 2
+* <math>(()) = 2</math>
-* ()() = ω
+* <math>()() = \omega</math>
 上述这些都跟ε(0)以下的tree相同
-* [] = ε(0)
+<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>
-* ([]) = ε(0)+1
-* (([])) = ε(0)+2
-* ([])() = ε(0)+ω
-* ([])([]) = ε(0)·2
-* (([]))([]) = ε(0)·3
-* (([])())([]) = ε(0)·ω
-* (([])([]))([]) = ε(0)^2
-* (([]))(([])) = ε(0)^ω
-* (([])([]))(([])([])) = ε(0)^ε(0)
-* []() = ε(1)
-* [](()) = ε(2)
-* [](()()) = ε(ω)
-* []([]) = ε(ε(0))
-* []([]([])) = ε(ε(ε(0)))
-* [()] = ζ(0)
-* ([()]) = ζ(0)+1
-* ([()])([()]) = ζ(0)·2
-* []([()]) = ε(ζ(0)+1)
-* [](([()])) = ε(ζ(0)+2)
-* []([]([()])) = ε(ε(ζ(0)+1))
-* [()]() = ζ(1)
-* [()]([()]) = ζ(ζ(0))
-* [(())] = ϑ(Ω·3) = φ(3,0)
-* []([(())]) = ϑ(Ω+ϑ(Ω·3))
-* [()]([(())]) = ϑ(Ω·2+ϑ(Ω·3))
-* [(())]() = ϑ(Ω·3+1)
-* [((()))] = ϑ(Ω·4)
-* [()()] = ϑ(Ω·ω)
-* [(()())] = ϑ(Ω·(ω+1))
-* [(()())(()())] = ϑ(Ω·ω^ω^ω)
-* [([])] = ϑ(Ω·ϑ(Ω))
-* [([])]() = ϑ(Ω·ϑ(Ω)+1)
-* [(([]))] = ϑ(Ω·(ϑ(Ω)+1))
-* [([])([])] = ϑ(Ω·ϑ(Ω)·2)
-* [([]())] = ϑ(Ω·ϑ(Ω+1))
-* [([()])] = ϑ(Ω·ϑ(Ω·2))
-* [([([])])] = ϑ(Ω·ϑ(Ω·ϑ(Ω)))
-* [[]] = ϑ(Ω^2) = Γ(0)
-* []([[]]) = ϑ(Ω+ϑ(Ω^2))
-* [()]([[]]) = ϑ(Ω·2+ϑ(Ω^2))
-* [([[]])] = ϑ(Ω·ϑ(Ω^2))
-* [([([[]])])] = ϑ(Ω·ϑ(Ω·ϑ(Ω^2)))
-* [[]]() = ϑ(Ω^2+1)
-* [[]]([[]]) = ϑ(Ω^2+ϑ(Ω^2))
-* [[]()] = ϑ(Ω^2+Ω)
-* [[]([[]])] = ϑ(Ω^2+Ω·ϑ(Ω^2))
-* <nowiki>[[()]]</nowiki>= ϑ(Ω^2·2)
-* [[([[]])]] = ϑ(Ω^2·ϑ(Ω^2))
-* [[[]]] = ϑ(Ω^3)
-* [[[]]]() = ϑ(Ω^3+1)
-* [[[]]()] = ϑ(Ω^3+Ω)
-* [[[]()]] = ϑ(Ω^3+Ω^2)
-* [[[()]]] = ϑ(Ω^3·2)
-* [[[[]]]] = ϑ(Ω^4)
-* [][] = ϑ(Ω^ω)
-* ([][]) = ϑ(Ω^ω)+1
-* []([][]) = ϑ(Ω+ϑ(Ω^ω))
-* [([][])] = ϑ(Ω·ϑ(Ω^ω))
-* [[([][])]] = ϑ(Ω^2·ϑ(Ω^ω))
-* [[][]] = ϑ(Ω^ω+1)
-* [[][]]() = ϑ(Ω^ω+2)
-* [[][]]([][]) = ϑ(Ω^ω+ϑ(Ω^ω))
-* [[[][]]] = ϑ(Ω^ω+Ω)
-* [[[][]]]() = ϑ(Ω^ω+Ω+1)
-* [[[][]]()] = ϑ(Ω^ω+Ω·2)
-* [[[[][]]]] = ϑ(Ω^ω+Ω^2)
-* [[[[[][]]]]] = ϑ(Ω^ω+Ω^3)
-* [()][] = ϑ(Ω^ω·2)
-* [[()][]] = ϑ(Ω^ω·2+1)
-* [[[()][]]] = ϑ(Ω^ω·2+Ω)
-* [[[[()][]]]] = ϑ(Ω^ω·2+Ω^2)
-* [(())][] = ϑ(Ω^ω·3)
-* [([][])][] = ϑ(Ω^ω·ϑ(Ω^ω))
-* [[]][] = ϑ(Ω^(ω+1))
-* [[[]]][] = ϑ(Ω^(ω+2))
-* [[][]][] = ϑ(Ω^(ω·2))
-* [[[][]][]] = ϑ(Ω^(ω·2)+1)
-* [[[[][]][]]] = ϑ(Ω^(ω·2)+Ω)
-* [[[][]]][] = ϑ(Ω^(ω·2)+Ω^ω)
-* [[[][]]()][] = ϑ(Ω^(ω·2)+Ω^ω·2)
-* [[[[][]]]][] = ϑ(Ω^(ω·2)+Ω^(ω+1))
-* [[()][]][] = ϑ(Ω^(ω·2)·2)
-* [[[]][]][] = ϑ(Ω^(ω·2+1))
-* [[[][]][]][] = ϑ(Ω^(ω·3))
-* [[[[][]][]][]][] = ϑ(Ω^(ω·4))
-* [()][()] = ϑ(Ω^ω^2)
-* [[()][()]] = ϑ(Ω^ω^2+1)
-* [[[()][()]][]] = ϑ(Ω^ω^2+Ω^ω)
-* [(())][()] = ϑ(Ω^ω^2·2)
-* [[]][()] = ϑ(Ω^(ω^2+1))
-* [[()][()]][()] = ϑ(Ω^(ω^2·2))
-* [(())][(())] = ϑ(Ω^ω^3)
-* [([])][([])] = ϑ(Ω^ϑ(Ω))
-* [[]][[]] = ϑ(Ω^Ω) = LVO
 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
@@ 第153行： / 第59行： @@
 * <nowiki>ψ(Ω₂^Ω₂^Ω₂)={{}}{{}}</nowiki>
 * ψ(Ω₂^Ω₂^Ω₂^ω)=<nowiki>{{}{}}</nowiki><nowiki>{{}{}}</nowiki>
+* ψ(Ω₃)=<>
+* ψ(Ω₄)=…
-于是得到ψ(Ω_ω)=极限
+于是得到ψ(Ω_ω)是wbtree的极限。
-{{默认排序:个人记号}}
+{{默认排序:相关问题}}
+[[分类:记号]]