Building Simple EV Trees by Hand

Assumptions: All examples assume 100bb effective stacks in a 6-max online cash game with no rake, unless a different stack size is stated in the example.

Everything in this module so far — calls, bluffs, value bets, line comparisons — has secretly been the same object: a small tree of outcomes with probabilities and payoffs at the ends. This lesson makes the object explicit and gives you a repeatable method you can run with a notepad in two minutes. Solvers build trees with millions of nodes; you'll build trees with four or five. The discipline is identical, and for one-street decisions, four or five branches is genuinely enough to get the answer right.

The method

For any single decision (we'll use "should I bet the river?"), five steps:

1. Draw the decision node. Your action, with its cost, at the root. Write down the pot.
2. Branch on every opponent response. Fold, call, raise — and split "call" into call by worse and call by better, because those leaves pay differently. Attach your estimated probability to each branch. They must sum to 100%. This sum check is half the value of the method: it forces "he folds a lot" and "he calls with everything worse" to fight for the same probability mass instead of both being vaguely true.
3. Write the payoff at each leaf, measured from the decision point: what do you collect from the pot, plus or minus the new money, if the hand ends down this branch?
4. Multiply each branch (probability × payoff) and sum. That's the line's EV.
5. Build the rival tree (here: check) the same way and compare. Then stress-test — more on that below.

The hand

BTN

7♦♦7♦

7♣♣7♣

BBunknown

Flop

A♠♠A♠

Turn

You open 7♦7♣ on the button, the big blind calls, and you check back your set on the A♠K♦7♥ flop — a trappy line on a board that smashes your range anyway. You bet the 2♣ turn small, he calls. The river 2♦ fills you up: sevens full of deuces. He checks. Pot 17.5bb — call it 18bb for clean numbers. You're considering 12bb.

Before the branches, do the range work in your head: he check-called turn on A-K-7-2, so he has lots of Ax and Kx one-pair hands (all worse than your full house), some floats that gave up, and a few monsters that beat you — slowplayed AA/KK from preflop, and 22 that just made quads. That inventory is what the probabilities below summarize. Note one trap this board sets: villain's A2 and K2 make deuces full, which your sevens full beats — fulls are ranked by their trips first. The only hands that beat you are AA, KK, and 22: seven combos.

Tree one: bet 12bb

Estimates, with the sum check shown:

• He folds: 40%. Busted floats and the weakest king-highs and ace-highs give up. Leaf payoff: you take the pot, +18.
• He calls with worse: 35%. Ax and Kx pay off a reasonable river bet often enough. Payoff: pot plus his call, +30.
• He calls with better: 20%. The slowplayed AA/KK/22 that check-called this whole way. Payoff: you lose your bet, −12.
• He raises: 5%. Handle this with a stated simple response: against this passive player a river check-raise is the pure nuts, and you fold. Payoff: −12 (your bet is gone; you pay nothing more). Folding a full house to a 5% branch may be wrong against other players — the point here is that the tree must say what you'll do so the leaf has a number.

Sum check: 40 + 35 + 20 + 5 = 100%. Now multiply back:

EV(bet 12) = 7.2 + 10.5 − 2.4 − 0.6 = +14.7bb

That's the whole machine. Four multiplications and a sum, and the line has a value.

Tree two: check back

BTN

7♦♦7♦

7♣♣7♣

BBunknown

Flop

A♠♠A♠

Turn

The rival line's tree is almost embarrassingly small, which is exactly why comparing trees is easy. You're the button: checking back ends the hand at showdown. One node, two leaves, and the probabilities come straight from the same range estimate (40% + 35% = 75% of his range is stuff you beat; 20% + 5% = 25% has you beat):

• He has worse: 75%. You win the pot: +18.
• He has better: 25%. You win nothing: 0.

EV(check) = 0.75 × 18 + 0.25 × 0 = +13.5bb

Comparison: bet +14.7 vs check +13.5 — betting is better by 1.2bb. Worth noticing where that 1.2bb lives: the fold branch pays the same as showdown (those hands lose to you either way), so the bet's entire edge is 12 × (0.35 worse-calls − 0.25 better-calls-plus-raises) = +1.2bb. The tree just re-derived the value-betting rule from the earlier lesson — that's the sign you built it right.

Stress-test before you trust it

Your probabilities are estimates, so the final step is asking: if my numbers are off, does the answer survive? Rerun the tree with the most uncertain input — fold% — shifted ten points each way. When fold% moves, that probability mass has to come from or go to somewhere; here, the hands whose response we're least sure about are the worse Ax/Kx (the better hands always continue), so the mass trades against call-by-worse:

Folds more (50% fold, 25% call worse, 20% call better, 5% raise): EV(bet) = 0.50 × 18 + 0.25 × 30 + 0.20 × (−12) + 0.05 × (−12) = 9 + 7.5 − 2.4 − 0.6 = +13.5bb — exactly ties checking. Makes sense: every worse hand that switches from calling to folding converts a +30 leaf into a +18 leaf, and at 25/25 the call columns cancel.

Folds less (30% fold, 45% call worse, rest unchanged): EV(bet) = 5.4 + 13.5 − 2.4 − 0.6 = +15.9bb — even better.

So across the whole plausible fold range, betting runs from "ties checking" to "clearly best." That's a robust decision: bet, and don't sweat the read. Contrast the input that would scare you: suppose your range estimate is wrong and villain has more slowplays — 30% call-by-better with worse-calls still 35% (his range is 30% better hands now, so checking changes too). Then EV(bet) = 5.4 + 10.5 − 3.6 − 0.6 = +11.7bb against EV(check) = 0.65 × 18 = +11.7bb — dead even, and any further slowplay-heaviness flips the bet negative. Your decision is robust to how often the worse hands call and fragile to how much of his range beats you. Knowing which input the answer hinges on is the real output of sensitivity work: it tells you what to watch for at the table, and which reads are worth paying for.

Why bother, when you'll never do this in-game

Three reasons this drill earns its place in your study routine:

• It catches double-counting. The most common EV error beginners make in their heads is letting villain fold 50%, call with worse 40%, and show up with better hands "pretty often" — probabilities that sum to far more than 100%. A written tree makes that impossible.
• It forces a plan for every branch. The 5% raise branch was worth −0.6bb because we decided in advance to fold. Players who never tree their spots face the check-raise with no plan and donate the pot-size mistake the tree priced at half a big blind.
• It builds the in-game shortcut. After twenty hand-built trees, "folds 40, worse calls 35, beats me 25 — bet small" runs in your head in five seconds. The notepad version is the training wheels for the instinct, and the instinct is just the tree compressed.

A note on scale: this was a river tree, so every leaf was terminal. Turn and flop trees sprout sub-branches (cards to come, future bets), and the honest way to handle that by hand is what the draw lesson did — collapse the future into a stated payoff or realized-equity estimate at each leaf. Keep your hand-built trees to one street and four or five branches. Past that, precision becomes false precision; your probability estimates carry ±10-point error bars, which is exactly why the sensitivity pass — not the second decimal — is what makes the answer trustworthy.

The checklist

Pin this above your review sessions:

1. Write the pot. Write your action and its cost.
2. Branch every response: fold / call-by-worse / call-by-better / raise. Probabilities sum to 100%.
3. Payoff at every leaf, measured from the decision point — including a stated plan for the raise branch.
4. Multiply, sum: one EV per line.
5. Tree the alternative line. Compare.
6. Shift the shakiest probability ±10 points. If the ranking holds, act with confidence; if it flips, identify the input it hinges on and go get a better read on exactly that.