The three patterns

In b1 you built one tree - GMV = Traffic x Conversion x AOV - and broke it in the simulator until naming the guilty driver was a reflex. That tree was multiplicative: a product of drivers. But not every metric multiplies. Revenue split by region is a sum. Active users this month is a balance carried from last month, a bridge. This session gives you the full vocabulary so that the moment a metric lands on your desk, you already know which shape it is and how its drop will read.

A multiplicative tree is a product of drivers, usually a volume times one or more rates times a value. Revenue = Users x ARPU. GMV = Traffic x Conversion x AOV. Its signature property - the one that makes it the friendliest tree to diagnose - is clean pass-through: a percentage change in any single driver moves the top line by that same percentage. Users down 10% with ARPU flat, revenue down 10%. Nothing gets absorbed.

{
  "unit": "$",
  "root": {
    "label": "Revenue", "op": "x",
    "children": [
      { "label": "Users", "value": 42000, "unit": "users" },
      { "label": "ARPU",  "value": 5, "unit": "$" }
    ]
  }
}

An additive tree is a sum of segments: total revenue = North America + EMEA + APAC. Regions, product lines, cohorts, channels - anything that partitions the whole. A drop here does not pass through evenly; it is usually one part falling while the others hold. To read it you need each part's weight (its share of the total), because a big swing in a tiny segment barely moves the top line, and a small swing in a dominant one moves it a lot.

{
  "unit": "$",
  "root": {
    "label": "Total revenue", "op": "+",
    "children": [
      { "label": "North America", "value": 120000, "unit": "$" },
      { "label": "EMEA",          "value": 62000,  "unit": "$" },
      { "label": "APAC",          "value": 38000,  "unit": "$" }
    ]
  }
}

Some metrics are stocks: they carry a balance forward. Active users, ARR, headcount - this month's number is last month's plus what came in minus what left. A bridge tree makes the flows explicit: Ending = Start + New + Resurrected - Churned. The whole point is that you watch gross adds against gross losses, not just the net. A flat active-user count can hide a business bleeding both ways fast - big adds papering over big churn.

{
  "unit": "users",
  "root": {
    "label": "Ending active users", "op": "bridge",
    "children": [
      { "label": "Start",       "value": 50000, "unit": "users" },
      { "label": "New",         "value": 8000,  "unit": "users" },
      { "label": "Resurrected", "value": 1500,  "unit": "users" },
      { "label": "Churned",     "value": 6000,  "unit": "users", "sign": -1 }
    ]
  }
}

Three live trees, one per pattern. For each: press Simulate a drop, follow the coral trail, then say out loud which pattern this is and which driver moved. The muscle you are building is instant classification - shape first, culprit second - because the shape tells you how to read the fall before you even look at the number.

{
  "unit": "$",
  "root": {
    "label": "GMV", "op": "x",
    "children": [
      { "label": "Traffic",    "value": 90000, "unit": "visits" },
      { "label": "Conversion", "value": 0.03,  "pct": true },
      { "label": "AOV",        "value": 55,    "unit": "$" }
    ]
  }
}

{
  "unit": "signups",
  "root": {
    "label": "Total signups", "op": "+",
    "children": [
      { "label": "Organic",  "value": 4200, "unit": "signups" },
      { "label": "Paid",     "value": 2800, "unit": "signups" },
      { "label": "Referral", "value": 1500, "unit": "signups" }
    ]
  }
}

{
  "unit": "$",
  "root": {
    "label": "Ending ARR", "op": "bridge",
    "children": [
      { "label": "Start",     "value": 1200000, "unit": "$" },
      { "label": "New",       "value": 240000,  "unit": "$" },
      { "label": "Expansion", "value": 90000,   "unit": "$" },
      { "label": "Churn",     "value": 150000,  "unit": "$", "sign": -1 }
    ]
  }
}

Now you drive. The two trees below are yours to edit - swap in a metric you actually own. Then answer the third question in prose. The goal is to leave able to look at any metric on your dashboard and instantly say "that is a product / a sum / a bridge."

{
  "unit": "$",
  "root": {
    "label": "Monthly revenue", "op": "x",
    "children": [
      { "label": "Leads",      "value": 800,  "unit": "leads" },
      { "label": "Close rate", "value": 0.22, "pct": true },
      { "label": "Deal size",  "value": 1200, "unit": "$" }
    ]
  }
}

{
  "unit": "$",
  "root": {
    "label": "Total revenue", "op": "+",
    "children": [
      { "label": "Product A", "value": 95000, "unit": "$" },
      { "label": "Product B", "value": 68000, "unit": "$" },
      { "label": "Product C", "value": 41000, "unit": "$" }
    ]
  }
}

• List your team's top five metrics. Beside each, write its shape: product, sum, or bridge. If you cannot decide, that metric is not yet understood - a useful discovery.
• For the sums, rank the parts by contribution. You should be able to say which two segments carry most of the total from memory.
• For any bridge, find where "adds" and "losses" live in your data - two different queries, two different owners. Most dashboards only show the net; that is the gap.
• Take one metric currently shown as a single number and re-draw it as its shape in the simulator. Paste your numbers into any tree above.
• Bring one bridge metric to b3 - we will hang a leading indicator on it and run the diagnosis playbook.

The three patterns are not one person's invention - they are the working core distilled from a handful of durable frameworks, applied here live instead of described. This page draws on: