IB Math AI AHL 3.16 - Tree and Cycle Algorithms

This IB Math AI HL 3.16 – Tree and Cycle Algorithms slide deck provides a comprehensive, algorithm-focused exploration of graph theory and its real-world optimization applications. The lesson builds from foundational definitions of walks, trails, paths, circuits, and cycles, establishing the language needed to analyze undirected and weighted graphs with precision.

Students learn to identify and test for Eulerian trails and circuits using vertex degree conditions, and to recognize Hamiltonian paths and cycles in network problems. The deck develops strong procedural understanding of minimum spanning trees through both Kruskal’s and Prim’s algorithms, emphasizing how and why edge selections minimize total weight in practical contexts such as wiring and infrastructure design.

Advanced applications include the Chinese Postman Problem and the Travelling Salesman Problem, where students apply structured algorithms to determine shortest routes and optimal cycles. Upper and lower bounds for TSP are explored using the Nearest Neighbor and Deleted Vertex algorithms, reinforcing strategic problem-solving and interpretation of bounds rather than brute-force enumeration.

With its clear visuals, step-by-step algorithms, and exam-style practice problems, this slide deck supports HL-level classroom instruction and independent study, equipping students with the analytical tools needed to confidently solve complex network optimization problems in IB Mathematics assessments.