演化樹(Evolutionary tree)不僅對於生物分類極為重要，更可藉由演化樹來得知生命的起源。我們將以DNA這個最原始最直接的資訊，略過生物的構造，嘗試建立和真正演化情況最相近的演化樹。
建構演化樹的基礎有三種類型：「特徵」(Character base)、「序列」(Sequence base)及「距離」(Distance base)。除了不同類型的基礎外，尚有多種不同計分函數。除了少數特殊計分函數配合特殊輸入資料以外，絕大多數的演化樹問題均為NP-hard。我們將以距離為基礎，設計建構演化樹的演算法。
我們將提出兩種建構演化樹的方法：
方法一：提出一演算法，並利用距離函數滿足三角不等式的特性，證明其近似解倍率之上限值不超過1.44 log n + 1。
方法二：提出一動態演算法，其在固定樹葉節點順序時，可建構最佳ultrametric演化樹。並尋找建構演化樹時所需要的較佳樹葉節點循環順序，再利用此順序建構演化樹。

In this thesis, we shall propose heuristic algorithms to construct evolutionary trees under the distance base model. For a distance matrix of any type, the problem of constructing a minimum ultrametric tree (MUT), whose scoring function is the minimum tree size, is NP-hard. Furthermore, the problem of constructing an approximate ultrametric tree with approximation error ratio within $n^{epsilon}, epsilon > 0$, is also NP-hard. When the distance matrix is metric, the problem is called the triangle minimum ultrametric tree problem ($ riangle$MUT). For the $ riangle$MUT, there is a previous approximation algorithm, with error ratio $leq 1.5 ( lceil log n
ceil + 1 )$. And we shall propose an improvement, with error ratio $leq lceil log_{alpha} n
ceil + 1 cong 1.44 lceil log n
ceil + 1$, where $alpha = frac{sqrt{5}+1}{2}$ for solving the $ riangle$MUT problem.

We shall also propose a heuristic algorithm to obtain a good leaf
node circular order. The heuristic algorithm
is based on the clustering scheme. And then we shall design a dynamic
programming algorithm to construct the optimal ultrametric tree with
some fixed leaf node circular order. The time complexity of the
dynamic programming is $O(n^3)$, if the scoring function is the
minimum tree size or $L^1$-min increment.

Chapter 1. Introduction
Chapter 2. Preliminaries
2.1 Distance Matrices and Evolutionary Trees
2.2 Optimization Problems
2.2.1 Scoring functions
2.2.2 Complexity
2.3 The Unweighted Pair Group Method with Maximum (UPGMM) for Ultrametric Trees
2.4 The Neighbor Joining Method for Additive Trees
2.5 An Approximation for riangle MUT
Chapter 3. An Approximation Algorithm for riangle MUT
3.1 The Binary Splitting Tree Problem
3.2 Our Approximation Algorithm
Chapter 4. A Heuristic Algorithm for MUT
4.1 The Leaf Node Circular Order
4.2 The Optimal Ultrametric Tree for a Certain Fixed Leaf Node Circular Order
Chapter 5. Experiment Results
Chapter 6. Conclusion

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