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Expanding/fixing my comment: I preface with history, Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of marked points). With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points. If there are no marked points, the only compactness phenomenon is energy concentration, and ghost bubbles can form betwixt energy-concentrated components if (for example) such ghosts have three nodal points.

Look at Lemma 4.2 in Parker-Wolfson and how it is used later on (pg. 91-92): Their ghost bubbles sit in a sequence which must terminate at a bubble/component having energy concentration, so the bubble tree is still finite. (I originally thought "unstable" ghost bubbles could arise in a chain, not contradicting finiteness of the tree, but nope!)

Restating, if a stable ghost bubble in a bubble tree has less than 3 marked points then: It is either at the end of a tree branch with 2 marked points (and 1 node), or it is in the middle of a chain with 1 or 2 marked points (and 2 nodes), or it is a ``connector'' for at least 3 other components (i.e. it has at least 3 nodes).

This is also discussed in Parker's AMS notice "What is... a Bubble Tree?".

Expanding/fixing my comment: I preface with history, Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of marked points). With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points. If there are no marked points, the only compactness phenomenon is energy concentration, and ghost bubbles can form betwixt energy-concentrated components if (for example) such ghosts have three nodal points.

Look at Lemma 4.2 in Parker-Wolfson and how it is used later on (pg. 91-92): Their ghost bubbles sit in a sequence which must terminate at a bubble/component having energy concentration, so the bubble tree is still finite. Note that if each step of the bubbling process produced infinitely many ghost bubbles then the curve (bubble tree) would be noncompact.

Restating, if a stable ghost bubble in a bubble tree has less than 3 marked points then: It is either at the end of a tree branch with 2 marked points (and 1 node), or it is in the middle of a chain with 1 or 2 marked points (and 2 nodes), or it is a ``connector'' for at least 3 other components (i.e. it has at least 3 nodes).

This is also discussed in Parker's AMS notice "What is... a Bubble Tree?".

Expanding/fixing my comment: I preface with history, Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of marked points). With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points. If there are no marked points, the only compactness phenomenon is energy concentration, and ghost bubbles can form betwixt energy-concentrated components if (for example) such ghosts have three nodal points.

Look at Lemma 4.2 in Parker-Wolfson and how it is used later on (pg. 91-92): Their ghost bubbles sit in a sequence which must terminate at a bubble/component having energy concentration, so the bubble tree is still finite. (I originally thought "unstable" ghost bubbles could arise, not contradicting finiteness of the tree, but nope.)

This is also discussed in Parker's AMS notice "What is... a Bubble Tree?".

Expanding/fixing my comment: I preface with history, Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of marked points). With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points. If there are no marked points, the only compactness phenomenon is energy concentration, and ghost bubbles can form betwixt energy-concentrated components if (for example) such ghosts have three nodal points.

Look at Lemma 4.2 in Parker-Wolfson and how it is used later on (pg. 91-92): Their ghost bubbles sit in a sequence which must terminate at a bubble/component having energy concentration, so the bubble tree is still finite. (I originally thought "unstable" ghost bubbles could arise in a chain, not contradicting finiteness of the tree, but nope!)

Restating, if a stable ghost bubble in a bubble tree has less than 3 marked points then: It is either at the end of a tree branch with 2 marked points (and 1 node), or it is in the middle of a chain with 1 or 2 marked points (and 2 nodes), or it is a ``connector'' for at least 3 other components (i.e. it has at least 3 nodes).

This is also discussed in Parker's AMS notice "What is... a Bubble Tree?".

Expanding/fixing my comment: Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of marked points). With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points. If there are no marked points, the only compactness phenomenon is energy concentration, and ghost bubbles can form betwixt energy-concentrated components if (for example) such ghosts have three nodal points.

Look at Lemma 4.2 (in Parker-Wolfson) and how it is used later on (pg. 91-92): Their ghost bubbles sit in a chain which must terminate at a bubble/component having energy concentration, so in their setup the bubble tree is still finite. In modern language, some of those ghost bubbles are non-stable.

Expanding/fixing my comment: I preface with history, Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of marked points). With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points. If there are no marked points, the only compactness phenomenon is energy concentration, and ghost bubbles can form betwixt energy-concentrated components if (for example) such ghosts have three nodal points.

Look at Lemma 4.2 in Parker-Wolfson and how it is used later on (pg. 91-92): Their ghost bubbles sit in a sequence which must terminate at a bubble/component having energy concentration, so the bubble tree is still finite. (I originally thought "unstable" ghost bubbles could arise, not contradicting finiteness of the tree, but nope.)

This is also discussed in Parker's AMS notice "What is... a Bubble Tree?".