生成樹在圖論中扮演著重要角色，提供了關於連通性與計數的見解。本論文研究如何在多種圖（例如扇形圖、輪子圖以及梯子圖）中計算生成樹數量，並採用了代數與組合兩種方法。在代數方法中，我們利用拉普拉斯矩陣的譜來獲取生成樹的數量；而在組合方法中，我們通過建立遞迴關係來推導其數量。此外，我們建立了生成樹計數與鋪磚問題之間的聯繫，這使得我們能夠輕鬆地推導出一個簡單的遞迴公式。
本論文還提出了 Cayley 公式的另一種證明方式，並研究了完全圖中生成樹的統計量。此外，我們採用了extended Prüfer code的概念來探究生成樹的更深層結構特性。我們還在Prüfer code構造過程中固定刪除節點的順序，進而計算其生成樹的數量。

Spanning trees play an important role in graph theory, providing valuable insights into connectivity and enumeration. This thesis studies the methods for counting spanning trees in various graphs, such as the fan graphs, the wheel graphs, and the ladder graphs, using both algebraic and combinatorial approaches. For algebraic approach, we use the spectrum of Laplacian matrix to acquire the number of spanning trees, as for the combinatorial approach, we derive the quantity by finding a recursive relation. We also establish connections between counting spanning trees and tiling problems, which easily give a simple recursive formula.

The thesis also presents an alternative proof of Cayley’s formula and investigates statistical properties of spanning trees in complete graphs. Additionally, we adopt the concept of extended Prüfer code to find out a deeper structural properties of trees. We also enumerate the number of spanning trees that have the same order of deleted leaves during the construction of Prüfer code.

[1] A. Benjamin and J. Quinn. Proofs that Really Count: The Art of Combinato-
rial Proof. Dolciani Mathematical Expositions, Mathematical Association of
America, 2003.
[2] A. Cayley. “A Theorem on Trees.” Quart. J. Math., vol. 23, pp. 376–378,
1889.
[3] A. E. Brouwer and W. H. Haemers. Spectra of Graphs. Springer, New York,
2012.
[4] D. Foata and J. Riordan. “Mappings of acyclic and parking functions.” Ae-
quationes Mathematicae, vol. 10, pp. 10–22, 1974.
[5] H. Pr¨ufer. “Neuer Beweis eines Satzes ¨uber Permutationen.” Arch. Math.
Phys., vol. 27, pp. 742–744, 1918.
[6] J. Fran¸con. “Acyclic and parking functions.” Journal of Combinatorial The-
ory, Series A, vol. 18, pp. 27–35, 1975.
[7] J. H. van Lint and R. M. Wilson. A Course in Combinatorics, 2nd ed. Cam-
bridge University Press, 2001.
[8] J. Riordan. “Ballots and trees.” Journal of Combinatorial Theory, vol. 6, pp.
408–411, 1969.
[9] J. Rukavicka. “On generalized Dyck paths.” The Electronic Journal of Com-
binatorics, vol. 18, 2011.
[10] J. S. Kim and D. Stanton. “The Combinatorics of Associated Laguerre Poly-
nomials.” Symmetry, Integrability and Geometry: Methods and Applications
(SIGMA), vol. 11, 2015.
[11] K. Ch. Das. “A Sharp Upper Bound for the Number of Spanning Trees of a
Graph.” Graphs and Combinatorics, vol. 23, no. 6, pp. 625-632, 2007.
[12] M. P. Sch¨utzenberger. “On an enumeration problem.” Journal of Combina-
torial Theory, vol. 4, no. 3, pp. 219–221, 1968.
[13] N. Biggs. Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1974.
[14] R. B. Bapat. Graphs and Matrices, 2nd ed. Springer, London, 2014.
[15] R. Brigham, R. Caron, P. Chinn, and R. Grimaldi. “A Tiling Scheme for
the Fibonacci Numbers.” The Journal of Recreational Mathematics, vol. 28,
1997.
[16] R. Diestel. Graph Theory, 5th ed. Springer, Berlin, 2017.
[17] R. E. Jamison. “Centers and medians of tree-like graphs.” Graphs and Com-
binatorics, vol. 24, no. 3, pp. 185–192, 2008.
[18] R. P. Lewis. “The number of spanning trees of a complete multipartite graph.”
Discrete Mathematics, vol. 197-198, pp. 537–541, 1999.