Interpretation of open balls with the p- adic metric

I am going to give a presentation about p-adic numbers. While studying what p-adic numbers even are, I got stuck at the geometric interpretation of the open balls. I have read many times that every point in those balls are considered to be the centrum of the balls. I do believe that one because I could easily follow the mathematical prove. I just cannot interpret it physically...

For example, let say we have the metric space Q_3 with the 3-adic metric. Then we can have a open ball of radius (1/3)^0. In that ball we have three identical disjoint open balls of radius (1/3). And in each of those balls we again have three open balls with radius (1/3)^2 etc. See the picture :D But then we can interpret those balls as a graph. The vertex in the middle is the ball with radius one. The vertices to the right and bottom left are the three balls with radius (1/3). AND NOW MY PROBLEM: I dont understand what the vertex on the top left is. Can someone please make this clear to me?

• I got it from this YouTube video: youtube.com/… At time 15:43, you can see the picture. This guy is also explaining why there is vertex on the left bottom. Unfortunately I still do not get him. @RobArthan – esmo Apr 9 '17 at 7:45
• In the tree representation every vertex has degree 4 if it forks by 3 "going down". We need an edge to connect "up to the top" – Henno Brandsma Apr 9 '17 at 9:35
• Aaaahhhhh now I do get it! Super. Thank you @HennoBrandsma – esmo Apr 9 '17 at 9:43

In any ultrametric space (so a metric space $(X,d)$ with the strengthened triangle inequality $d(x,z) \le \max(d(x,y), d(y,z))$ for all $x,y,z$) any point $y$ for an open ball $B(x,r)$ is its "centre": $B(y,r) = B(x,r)$:
$z \in B(y,r)$, then $d(x,z) \le \max(d(y,z), d(x,y)) < r$, as $d(x,y) < r$ from $y \in B(x, r)$, and $d(y,z) < r$ from $z \in B(y,r)$. So $z \in B(x,r)$. The proof of the reverse: $z \in B(x,r)$ is in $B(y,r)$ is entirely similar. That makes it harder to interpret these balls geometrically, I think.