What used to be the hunting grounds for noblemen of the Holy Roman Empire was turned into a huge park in the 19th century (without losing any of its size). The Tiergarten is the green lung in the middle of the city and there's more than enough space for everyone. And yes: Most of former Westberlin is on its other side.
Apart from enjoying all that green, there’s a lot to discover at Tiergarten, like a path lined with gas lanterns from all over Europe; sculptures and monuments scattered throughout the park, and in the far western part you can look at animals in the Berlin zoo.
End your hike with a beer in the Schleusenkrug in the far western end. There's also lots of maps of the area inside the park so you can't get lost.