More than a family diagram
A coaching tree is usually presented as a chart: a head coach at the top, assistants branching beneath him, and their own assistants branching further still. The diagram is tidy and the reality behind it is not, because what actually spreads between those names is a mixture of scheme, organisational habits and personal trust built over years in the same building.
Two assistants who worked under the same coordinator can leave with different lessons entirely, depending on what they were responsible for and how closely they worked with him. The tree implies a single inheritance; the truth is closer to several people taking different notes in the same room.
Ideas get adapted, not copied
When an assistant becomes a head coach or coordinator elsewhere, he rarely installs his old boss's system unchanged. He installs a version filtered through the roster he now has, the staff he has been able to hire, and the parts of the original system he actually believed in rather than merely executed. A tree branch is a translation, not a photocopy.
This is why two coaches from the same lineage can look strikingly different on Sundays despite sharing a vocabulary. The vocabulary travels more easily than the judgement behind when to use it, and judgement is the harder thing to teach.
Why the metaphor persists anyway
Despite its flaws, the tree metaphor survives because hiring in the league leans heavily on trusted relationships, and those relationships really do cluster along lines that resemble branches. A general manager hiring a young coordinator is often betting on a pedigree because it is the clearest signal available, even if it is an imperfect one.
The honest way to use the idea is as a starting hypothesis rather than a conclusion: a coach's background tells you where to start looking for influences, not what you will necessarily find. Arena's NFL roster guide covers how staffs and personnel departments are actually structured underneath the head coach.