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Having text labels inside nodes in trees drawn by tikz-qtree is straightforward, but is it possible to also have a label outside the node? For example, a node with letter A inside it, and letter B outside it on its left/right-hand-side?

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You can add labels to nodes, which are nodes themselves, such as in this tikz-qtree line:

[.\node [label=right:B] {A};]

There are further placement options, such as left, above, below, above left, ... and style options, such as label={[red]right:B}.

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