在這篇論文中，首先介紹Galton-Watson分支過程以及多型態Galton-Watson分支過程的一些重要定義與命題。

接著，為了透過多型態Galton-Watson分支過程研究家庭結構的演化，我們提出「親子結構」的概念。親子結構是一種二層次的有根樹，用以表示不同類型個體所產生後代的特定模式。基於此概念，我們建構了一個新的隨機過程，稱為「親子結構過程」，以探討不同親子結構在族群所占比例之變化與預測其極限比例。我們證明此引發過程本身為一個多型態Galton-Watson 分支過程，因此，在非奇異性、正規性與超臨界等條件下，各種親子結構所佔的比例會幾乎必然收斂至一個向量中對應的分量，而此向量是透過將歸一化左特徵向量除以其分量總和後所得。

最後，我們透過數值模擬驗證在這篇論文中提出來的例子，得到數值模擬與理論預測呈現一致之結果。

本研究提供了一種新穎的機率方法，用以建模及理解複雜的家庭演化模式，並且可作為日後在人口統計學、社會科學以及族群建模等相關領域研究的基礎。

In this thesis, first, we give an introduction to some important definitions and propositions of the Galton-Watson branching processes and the multitype Galton-Watson branching processes.

Next, in order to study the evolution of family structures through multitype Galton-Watson branching processes, we present the concept of parent-child structures which are two-level rooted trees representing specific patterns of producing offspring of individuals of various types, and therefore, we construct a new stochastic process, called the parent-child structure process, to support us to predict how the proportion of each parent-child structure varies in the future. Also, we show that this induced process is a multitype Galton-Watson branching process, so under the conditions of nonsingularity, positive regularity, and supercritical case, the proportion of each parent-child structure converges almost surely to the corresponding component of a vector which is obtained by dividing the normalized left eigenvector by the sum of its components.

Finally, we conduct some numerical simulations of the example mentioned in this thesis to validate if the simulated outcome coincides with theoretical prediction.

This study provides a probabilistic approach to modeling and understanding complex family evolution patterns in a novel way and may serve as a foundation for future research in demography, social science, and population modeling.

[1] Krishna B. Athreya and Peter E. Ney. Branching processes. Courier Corporation, 2004.
[2] Peter L. Davies. The simple branching process: a note on convergence when the mean is infinite. Journal of Applied Probability, 15(3):466–480, 1978.
[3] Theodore E. Harris. The theory of branching processes. Springer Berlin, 1963.