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I am trying to practice with blender by creating a plant with bulb or bean shaped leaves as in this illustration (author unknown):

enter image description here

I created a leaf and used a Hair particle system to duplicate it around a trunk, but am having a hard time ordering the leaves nicely.

It doesn't have to be exactly as in the illustration, but at least somewhat regular. This is what I have gotten so far:

enter image description here

It's not great because the meshes of the leaves intersect and have irregular gaps between them. I tweaked around for a long time but none of my settings are set to "random".

How are the leaves emitted in this scene and how can I order them?

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Your particle setting is almost good. For 1 particle per face to work properly though you need to have as much or more particles as faces. Your emitter is pretty dense. I would suggest to lower the face number. Maybe try to make stem from 16 verts circle. I would also suggest to change Initial Orientation setting to Normal instead of Normal-Tangent to avoid different rotation on different rows of faces. Then play with size and rotation of particles until you get satisfactory result.Here is your file with the changes I mentioned.I think it's a good start for further experimentation:

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