Discrete Mathematics for BSAI

01 · Opening Scenario: Follow a Decision or Explore Them All?

A student-support guide asks whether a learner needs help now. If yes, it asks one follow-up question and suggests support. If no, it asks a different question and offers a learning resource. A learner follows one branch at a time. But how could an editor systematically visit every question and outcome in the guide?

Predict: Following a learner's answers may reach only one final action. Checking the whole guide requires visiting all branches. A rooted tree represents the branches; a traversal specifies the order in which we visit every node.

02 · Learning Objectives and Three-Hour Guide

By the end of Week 15, you should be able to identify a root, parent, child, leaf and depth; draw a small rooted binary tree from stated decisions; trace preorder, inorder, postorder and level order; distinguish one decision path from a complete traversal; and explain what different visit orders accomplish.

Classroom essentials Sections 01–12 and Practice 1–6. Optional self-study Colab and Practice 7. The visual step-through is a check after a paper trace.

03 · A Root Gives Direction to the Branches

A tree is a connected graph with no cycle. Choose one vertex as the root; it is the starting point for describing downward branches. In the diagram below A is the root. B and C are its children, while A is their parent. D and E are children of B; F and G are children of C.

A leaf has no children: D, E, F and G are leaves. Depth is the number of edges from the root to a node. For example, A→B→E uses two edges, so E has depth 2. The subtree at B consists of B and its descendants D and E.

04 · Left and Right Matter in a Binary Tree

A binary tree has at most two children at each node, identified as left and right. In Figure 1, B is the left child of A and C the right child. D is left of B, E right of B; F is left of C, G right of C.

The left/right positions are part of this tree's information. If we swap B and C together with their subtrees, the shapes may look similar, but the left-before-right traversal results change. A childless leaf has neither a left nor a right child.

05 · Following Answers Visits One Path

Suppose A asks “Needs support now?” A yes answer moves left to B, which asks “Missed several checkpoints?” A no answer at B moves right to E, the study-session suggestion. The decision path is A → B → E. We did not visit C, D, F or G.

A decision tree models a specified series of questions and outcomes. It does not automatically learn from data, establish that a recommendation works, or know unstated facts. Its branch labels and rules must be supplied and checked by people.

06 · A Traversal Visits Every Node

A traversal is an ordered visit of the entire tree. We will use four orders on the same seven-node tree. The letters in a visit list describe when each node is recorded; moving through the drawing to reach another node does not add an extra recorded visit.

For the first three orders, apply the same instruction within each smaller subtree. Level order proceeds across a whole depth before moving down.

07 · Preorder: Node Before Its Children

Begin at A and record it. Then finish the left subtree rooted at B: record B, D, E. Finally finish the right subtree rooted at C: record C, F, G.

Preorder is useful when we want to see each parent before its descendants, such as presenting the outline of a branching guide from the top.

08 · Inorder: Root Between Left and Right

Inorder is defined here for a binary tree with identified left and right children. First traverse A's left subtree: D, B, E. Record A. Then traverse A's right subtree: F, C, G.

At B, the instruction “left, node, right” becomes D, B, E. At C it becomes F, C, G. A leaf is recorded when reached because it has no child subtrees to visit first.

09 · Postorder: Children Before Parent

Finish B's children D and E before recording B; finish C's children F and G before recording C; then record the root A.

This order makes the parent appear after both of its subtrees. For example, when reviewing a guide from its outcomes back to its earlier questions, one could inspect the leaf outcomes before recording their parent question. The traversal order alone does not determine whether the guide is correct.

10 · Level Order: Across Each Depth

Record A at depth 0. Then B and C at depth 1, left to right. Then D, E, F and G at depth 2, left to right.

Level order differs from preorder: after B, level order visits C before going down to D. This is useful when we want to inspect all questions at one depth before inspecting deeper outcomes.

11 · Traversal Explorer

Predict → Test → Explain First write the full visit list on paper. Choose an order, predict the next recorded node, and check it. The diagram highlights nodes as they are recorded. A wrong guess leaves the step available for another try.

12 · Smaller Case Study 5: Exploring a Decision Tree

Scenario → Understand → Represent → Reason → Test → Interpret. A student-support guide has three yes/no questions and four possible actions. The seven nodes are the same ones in Figure 1, now with specific meanings:

Follow one learner

The learner answers yes at A and no at B. The path is A → B → E, leading to a study-session suggestion. No statement about C or its children is used for this learner. If the learner instead answers no at A and yes at C, the path is A → C → F.

Explore the whole guide

An editor may need to inspect every node. Preorder records a parent question before its branches: A,B,D,E,C,F,G. Level order examines the top question, then both second-level questions, then all four actions: A,B,C,D,E,F,G. Both visit all seven nodes; they differ in when C is inspected.

Interpret the model

The tree describes only the questions, answers and actions supplied here. It does not infer a student's needs from data or learn better rules. The action is a transparent recommendation under stated answers, not a diagnosis or proof of effectiveness.

13 · Optional Colab: Check One Hand Trace

Self-study · mathematics first Finish the paper preorder trace before running code. The small nested representation below stores each node as (label, left child, right child). None means that a child is absent. Parentheses group those three pieces; the whole structure follows the drawing.

tree = (
    "A",
    ("B", ("D", None, None), ("E", None, None)),
    ("C", ("F", None, None), ("G", None, None))
)

The next function reuses the recursive idea from Week 10. It stops at None. Otherwise, it separates one node into its label, left child and right child, prints the label first, and then visits the two children in that order.

def preorder(node):
    if node is None:
        return
    label, left, right = node
    print(label)
    preorder(left)
    preorder(right)

preorder(tree)

Predict A, B, D, E, C, F, G on separate lines. return at an absent child is the stopping case. This small check prints the preorder only; the other three orders remain hand-trace exercises first.

14 · Practice and Exit Check

Draw or point to the stated tree before writing each order. Selected solutions are at the end.

1. For Figure 1, give the root, B's parent and children, the leaves, and the depths of A, C and E.
2. Write preorder, inorder, postorder and level order of the Figure 1 tree. Underline the root in each list.
3. In the case study, a learner answers no at A and yes at C. Give the path and outcome. Is that path a traversal of the whole tree?
4. A different binary tree has root P; left child Q and right child R; Q has only a left child S; R has left child T and right child U. Draw it and find all four traversal orders.
5. Explain why inorder requires identified left and right children. If B and C swap positions with their subtrees in Figure 1, which preorder comes next?
6. An editor wishes to inspect all depth-1 questions before any leaf actions. Which traversal suits that aim? Show its list on Figure 1.
7. A learner says “Inorder is always alphabetical, because it lists D before B.” Identify the missing assumption and give the actual inorder list for Figure 1.

15 · Common Mistakes

16 · Summary

• A rooted tree has a chosen starting vertex. Parent/child relations follow outward from it; leaves have no children, and depth counts edges from root.
• A binary tree distinguishes left and right children. Preorder is node-left-right; inorder is left-node-right; postorder is left-right-node.
• Level order visits all nodes at each depth from left to right before going deeper.
• One learner's answer path reaches one action; a traversal records every node once.
• The case-study guide applies stated questions and branches. Its structure does not learn from data.

17 · Visual Concept Map

18 · Looking Ahead

Week 16 is integrated revision and final-examination preparation. We will compare models and reasoning methods already taught—logic, proofs, sets, sequences, counting, graph routes and tree traversals—without introducing new mathematical material.