Is there always one unique way to partition a sentence into its constituents or, like in the following picture, is it possible to analyze a sentence into parts several different ways?
The three rows of dots represent the same sentence, and the red underlines represent constituents, such as a noun phrase, compound verb, or prepositional phrase.
The way one analyses a sentence and divides it into parts depends on the system of analysis one uses. The sentence that follows, for example, can be divided in several ways:
The old man hit the young girl with a thin cane.
[The old man] [hit the young girl with a thin cane.]
[The old man] [[hit] [the young girl with a thin cane.]]
[The old man] [[hit] [the young girl {with a thin cane}.]]
[The old man] [[hit] [the young girl] {/with/ a thin cane}.]]
... and so on.
The answer to your question is decidedly NO -- there is not always a unique way to partition a sentence into its constituents -- and this is so however one defines "constituent." This is so because some sentences are inherently ambiguous: they have more than one syntactic reading, and so can be represented by different sets of constituents. The example used in the other answer is a case in point: "The old man hit the girl with a cane."
The ambiguity: did the man have the cane or did the girl? This is not specified in the sentence. (We may happen to know the answer, but it does not come from syntactic information. It might be found elsewhere in the discourse, or it might be based on our pragmatic knowledge -- old men are more likely to have canes than girls, etc. -- but it is not coded in this sentence itself.)
Two classic kinds of syntactic theories are phrase-structure grammars (PSG) and dependency grammars (DG). Both represent the sentence as a tree. In PSG, the internal nodes of the tree name "constituents", and one leaf of the tree corresponds to one word in the sentence. In DG each node in the tree corresponds to a word in the sentence, and we say that each node "depends" on its immediate parent node (i.e., the child depends on its parent).
Constituents: we must define what we mean by that -- the more appropriate way to analyze the sentence is a tree. Given a tree, we can speak of constituents by equating the notion of a constituent with the notion of a subtree. So if we say that each subtree forms one constituent, then your question makes sense.
Finally, note, that constituents are always nested, so your picture is oversimplified --- it shows no nesting.
Now, independently of y/our preferred theory, one could see our example as having constituents:
A. [ The old man ] [ hit [ the girl [ with a cane ] ] ].
or
B. [ The old man ] [ hit [ the girl ] [ with a cane ] ].
In A, [ with a cane ] modifies the GIRL (e.g., in DP terms, it depends on the node GIRL). It tells us which girl was hit: the one with the cane, and not the one with the ice-cream.
In B, [ with a cane ] modifies HIT (in DP terms, it depends on HIT), and tells us exactly how she was hit: with a cane, and not with an oxygen bag.
This kind of ambiguity (and many others) is what makes analysis of natural language challenging --- not so for computer languages, where ambiguity is absent by design.
