Trees as Recursive Structures

Week 8, Monday

February 23, 2026

PACING OVERVIEW (80 min total)

Part 1: Wordle Hook (10 min)

• Play 1-2 rounds of Wordle as a class
• Brief discussion: what makes a good guess?
• Introduce the idea that strategy = decision tree

Part 2: Jotto Activity (25 min)

• Explain rules and scoring (5 min)
• Hand out card decks, groups design strategies (15 min)
• Debrief: what first guess did you choose? why? (5 min)

Part 3: Tree Definitions (15 min)

• “Why Trees?” slide - quick tour of applications
• Knuth quote + “what’s special?” (connected, acyclic)
• Tree Terminology discovery questions (students answer)
• Vocabulary reference slide (skim - definitions are intuitive)

Part 4: Binary Trees & Recursion (20 min)

• Binary tree definition and recursive structure
• Height/node count relationship
• Why height matters for efficiency
• Connect back to Wordle/Jotto: height = worst-case guesses

Part 5: Wrap-up (10 min)

• Back to Wordle: decision tree perspective
• Summary slide
• Preview BSTs (Wednesday)

Let’s Play Wordle!

I’m thinking of a 5-letter word.

You have 6 guesses.

After each guess, I’ll tell you:

• 🟩 Green: Right letter, right position
• 🟨 Yellow: Right letter, wrong position
• ⬜ Gray: Letter not in word

Let’s play as a class!

Actually play 2-3 rounds with the class. Pick words ahead of time. Have students suggest guesses, vote on which to try. Track remaining possibilities on the board. Good words to use: CRANE, SLATE, MOIST, CHARM, PLUMB (words with common letters for game 1, then trickier ones).

If you have internet, you can play the actual game. Otherwise, run it manually on the board.

What Strategy Did We Use?

Think about how we played:

• What made a guess “good”?
• How did we narrow down possibilities?
• Did we have a system, or was it intuition?

The Strategy Question

Every guess splits the remaining possibilities.

Guess	Feedback	Words eliminated
CRANE	⬜🟨⬜⬜🟩	Most words without R and E in those spots
STORE	🟩⬜🟩🟨🟩	Now we know S, O, E positions + R somewhere
…	…	…

Good guesses split possibilities evenly.

Visualizing the Strategy

What if we drew every possible game?

                         SALET (first guess)
                        /       |        \        ...
                 🟩⬜⬜⬜⬜    🟨⬜⬜⬜⬜    🟩🟩⬜⬜⬜     ...
                    /           |            \
      SHIRK, SHOWY, SPICY   CRISP, MUSIC   SAPPY, SANDY
      SWORN, SNOWY, ...     PRISM, ...     SAUCY, ...

Each follow-up guess is consistent with prior feedback.

Each path from top to bottom is one game. This is a tree!

The Wordle Decision Tree

This structure has a name: a decision tree.

• Root: First guess
• Branches: Possible feedback patterns (there are _______)
• Internal nodes: Follow-up guesses
• Leaves: The answer (game over!)

The height of this tree = worst-case guesses needed.

The Optimal Strategy Question

Question: What first guess gives the best tree?

What does best mean?

Research answer: SALET, REAST, CRATE, TRACE

3-Letter Jotto: Let’s Build a Strategy

The setup:

• There are 16 possible secret words
• You guess a word; the response is the count of letters in common (0, 1, 2, or 3)
• You can only guess words that are still consistent with previous responses

Your task: Design a strategy — an algorithm that tells you what to guess in every situation, for any secret word.

Specify your strategy:

1. What’s your first guess?
2. For each possible response (0, 1, 2, or 3), what do you guess next?
3. And so on, until you’ve identified the secret word.

Jotto Scoring Examples

Secret	Guess	Score	Why
FOX	JOG	1	O matches
DEW	YAK	0	No letters in common
NIP	INK	2	I and N match
WOO	MOB	1	O matches (counted once)

The 16 Words

DEW  FOX  GEL  INK
JOG  MOB  NAP  NIP
NOR  NUT  OAT  OIL
PUG  SOY  WOO  YAK

Example: First Guess NIP

Each pile contains words consistent with that score.

Hand out card decks. Give students 5-7 minutes to play a few rounds, then regroup to discuss strategy.

Key questions to seed: - What’s a good first guess? Why? - Did anyone try guessing similar words in a row (CAT, BAT, HAT)? How did that go? - What’s the worst case — how many guesses could it take?

What Strategy Did You Discover?

Discussion questions:

• What makes a “good” first guess?
• What happens if you guess CAT, then BAT, then HAT?
• Can you always win in 2 guesses? Why or why not?

Lead into: the strategy IS a tree! Each guess partitions the remaining words by their Jotto score (0, 1, 2, or 3). Good guesses split evenly. Bad guesses leave big partitions.

Lower bound discussion: With 4 outcomes per guess and 16 words, theoretically log₄(16) = 2 guesses should suffice. But no word achieves a perfect 4-way split!

Why Trees?

Trees appear everywhere in computer science:

Domain	Tree Structure
File systems	Directories contain subdirectories
HTML/DOM	Elements contain child elements
Decision making	Each choice leads to sub-choices
Classification	“Is it X? If yes, go left…”
Game AI	Possible moves branch into more moves

Trees: The Most Important Nonlinear Structure

“Trees are the most important nonlinear structure in computer science.”

— Donald Knuth, The Art of Computer Programming

What’s special about this structure? ________________________

Tree Terminology

• One of the vertices is called the root of the tree. Guess which one it is.
• Make an English word containing the names of the vertices that have a parent but no sibling.
• How many parents does each vertex have?
• Which vertex has the fewest children?
• Which vertex has the most ancestors? descendants?
• What is d’s depth? What is d’s height?
• List all the vertices in b’s left subtree.
• List all the leaves in the tree.

Tree Vocabulary (reference)

• root: the single node with no parent
• leaf: a node with no children
• child: a node pointed to by me
• parent: the node that points to me
• sibling: another child of my parent
• ancestor: my parent or my parent’s ancestor
• descendant: my child or my child’s descendant
• subtree: a node and its descendants
• depth of node x: number of edges on path from root to x
• height of node x: number of edges on longest path from x to a leaf

d-ary Tree Definition

A d-ary tree T is either:

• _______________________________________

OR

• _______________________________________

Fill in via annotation: - Empty (no nodes) - A root node with up to d subtrees (each a d-ary tree)

Binary tree = d-ary tree with d=2

The tree shown has d=3 (ternary). Each subtree of the root is colored differently to emphasize the recursive structure.