Chinese (Mandarin) translation of my question may be an example of the sentence structure that both English and Chinese share:
[Is there any] [sentence structure] (that) [English has but Chinese (Mandarin) does not]? [有没有什么][句子结构][英语有但是汉语（普通话）没有]？
Consider preposition stranding, which is possible in English but not in Mandarin:
the girlfriend I broke up with
*gen fenshou de nvpengyou with breakup DE girlfriend
A resumptive pronoun is obligatory in Mandarin but impossible in English:
*the girlfriend I broke up with her
gen ta fenshou de nvpengyou with her breakup DE girlfriend
There are several other cases such that extraction out of a constituent is grammatical in English but not in Mandarin. In other words, many phrases in Mandarin including 把+NP and the aforementioned prepositional phrases seem to be extraction islands.
EDIT: I can also think of a case of the converse (a Mandarin syntactic construction which cannot be rendered in English without some circumlocution).
As jogloran pointed out, preposition stranding (and how you can't do it in Chinese) is a good source of these. Here are several examples of relative clauses where the object of a preposition is being pulled out. Note that the Chinese grammar doesn't match the English (because the obvious way to do it is ungrammatical):
carry a bag for X
I'm carrying this bag for him.
我帮他拿这个包。 I help him grab this CL bag
The person I carried this bag for is gone.
我帮他拿这个包的人走了。 I help him grab this CL bag REL person leave PFV
come back from X
I just came back from France.
我刚从法国回来。 I just from France return
The country I just came back from uses the Euro.
我刚从那里回来的国家用欧元。I just from there return REL country use Euro
Literally hit a phone for X
I called Apple (the computer company).
我给苹果公司打了电话。 I give apple company hit PFV telephone
The company I called makes computers.
我打了电话的公司是制造电脑的。 I hit PFV telephone REL company is make computer
poke your arm with X
I poked your arm with a chopstick.
我用筷子戳你的胳膊。 I use chopstick poke you POS arm
The chopstick I poked your arm with is on the floor.
我用来戳你胳膊的筷子在地上。 I use-come poke you (POS) arm REL chopstick located floor-on
sell my bike to X
I sold him my bike.
我给他卖我的自行车。 I give him sell I POS bike
The person I sold my bike to also lives in China.
我把自行车卖给(他?)的那个人也住在中国。I OBJ-Marker bike sell give (him?) REL that CL person also live in China
It seems there are a few different grammatical strategies:
Delete the preposition (as in #3); the meaning is still clear from the context.
Use a "dummy variable" as the object of the preposition (
他 = he in #1 and #5,
那里 = there in #2)
Reverbalization of the preposition using
来 = come (e.g., #4)
Notes on glosses:
CL = Classifier, as in the measure words in Chinese (e.g., 个).
REL = Relativizer; the 的 particle which links adjectives or relative clauses to nouns.
PFV = Perfective; the 了 particle which marks for completedness of verbs.
English participial phrases and absolute clauses come to mind.
Take this sentence in English, for instance:
Seeing the yellow light, I slowed down.
The Mandarin equivalent would need to be something like this:
[Wǒ] [kàn-dào huángdēng de shíhòu] [màn le xiàlai]. I see-EXTENT-complete yellow light MOD time slow ASP-perf EXTENT-down
It could be loosely stated as two uncoordinated verbal phrases, but they would be more disconnected than the English construction.
[Wǒ] [kàn-dào huángdēng], [wǒ] [màn le xiàlai]. I see-EXTENT-complete yellow light I slow ASP-perf EXTENT-down
Which is better rendered in English as 'I saw the yellow light; I slowed down.'
A sentence using an absolute construction in English, on the other hand, does not always make sense uncoordinated in Mandarin.
Weather permitting, I will see you tomorrow.
The conditional needs to be stated in Mandarin:
[Tiānqì] [yǔnxǔ de huà], [míngtiān jiàn]. weather permit if tomorrow see
Which is more like "In the case that weather permits, I will see you tomorrow" in English.
Notes on glosses:
MOD - modifier; makes the preceding phrase modify the following phrase
EXTENT - extent marker; -complete indicates completion, and -down indicates gradual completion.
ASP - aspect marker; -perf indicates perfective.