fallen
to fall
To move from top to bottom due to gravity; to fall to the ground.
Vorsicht, die Blätter fallen von den Bäumen und der Weg ist rutschig.
Careful, the leaves are falling from the trees and the path is slippery.

to fall
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to fall
To move from top to bottom due to gravity; to fall to the ground.
Vorsicht, die Blätter fallen von den Bäumen und der Weg ist rutschig.
Careful, the leaves are falling from the trees and the path is slippery.

The verb 'fallen' (to fall) always forms compound tenses (such as the perfect tense) with the auxiliary verb 'sein' (to be). This is because it describes movement or a change of state.
Der teure Teller ist mir aus der Hand gefallen.
The expensive plate fell out of my hand.
The past participle of 'fallen' (to fall) is 'gefallen' (fallen). This word is identical to the infinitive of the verb 'gefallen' (to please). You distinguish them by the auxiliary verb: 'Der Apfel ist gefallen' (The apple has fallen; fallen, with 'sein'), but 'Das Geschenk hat gefallen' (The gift was pleasing; gefallen, with 'haben').
Der Schnee ist leise gefallen.
The snow fell quietly.
Whether 'fallen' (to fall) is meant literally (to fall down) or figuratively (to decrease) can be recognized from the subject of the sentence. If it involves abstract values like 'Preise' (prices), 'Temperaturen' (temperatures), or 'Kurse' (rates/share prices), 'fallen' always means 'sinken' (to sink/decrease) or 'abnehmen' (to decline).
Der Aktienkurs der Firma ist plötzlich gefallen.
The company's share price suddenly fell.
In the expression 'jemandem schwer/leicht fallen' (to be hard/easy for someone), the person who experiences something as difficult or easy is always in the dative. The subject is often the impersonal 'es' (it).
Es fällt ihm schwer, früh aufzustehen.
It is hard for him to get up early.
The meaning 'to be killed in war' is very specific. The word 'gefallen' (fallen) is used in this context as a formal, almost euphemistic expression for the death of soldiers.
An dem Denkmal stehen die Namen der gefallenen Soldaten.
The names of the fallen soldiers are written on the memorial.
In a passive-like meaning, 'fallen' (to be made/said) is used with impersonal subjects like 'Entscheidung' (decision), 'Urteil' (judgment), or 'Wort' (word). It means that something is decided or said.
In der Besprechung ist noch keine Entscheidung gefallen.
No decision has been made in the meeting yet.
to lie / be located
'liegen' and 'fallen' can both have to do with a position on the ground. But the difference is important: 'fallen' describes the downward movement, often sudden or unwanted. 'liegen' describes the state afterward, or generally a horizontal position. Learners may confuse the words because after a fall, someone is lying on the ground. In practical terms: first you fall, then you may be lying on the ground.

Auf der nassen Treppe ist Anna fast gefallen.
Anna almost fell on the wet stairs.
Nach dem Training liegt Max müde auf dem Sofa.
After training, Max lies tired on the sofa.